INTERNATIONAL GONGRESS ive-mml>, thus the scheme-theoretical intersection Spec ( Q δ c 2 ω 1 ) Spec ⁡ Q δ c 2 ω 1 Spec(Q_(delta_(c)^(2omega_(1))))\operatorname{Spec}\left(Q_{\delta_{c}^{2 \omega_{1}}}\right)Spec⁡(Qδc2ω1) of the upward flow W δ c 2 ω 1 + W δ c 2 ω 1 + W_(delta_(c)^(2omega_(1)))^(+)W_{\delta_{c}^{2 \omega_{1}}}^{+}Wδc2ω1+and the nilpotent cone h 1 ( 0 ) h − 1 ( 0 ) h^(-1)(0)h^{-1}(0)h−1(0) is the line ( a 0 ) a 0 (a_(0))\left(a_{0}\right)(a0) with a double embedded point at the origin. Note that this upward flow was studied in [29, $8.2].
For d = 3 d = 3 d=3d=3d=3, we have the multiplicity-2 algebra
(3.11) Q δ c 3 ω 1 C [ a 0 , a 1 , a 2 ] / ( a 0 2 , a 0 a 1 , a 0 a 2 + a 1 2 ) (3.11) Q δ c 3 ω 1 ≅ C a 0 , a 1 , a 2 / a 0 2 , a 0 a 1 , a 0 a 2 + a 1 2 {:(3.11)Q_(delta_(c)^(3omega_(1)))~=C[a_(0),a_(1),a_(2)]//(a_(0)^(2),a_(0)a_(1),a_(0)a_(2)+a_(1)^(2)):}\begin{equation*} Q_{\delta_{c}^{3 \omega_{1}}} \cong \mathbb{C}\left[a_{0}, a_{1}, a_{2}\right] /\left(a_{0}^{2}, a_{0} a_{1}, a_{0} a_{2}+a_{1}^{2}\right) \tag{3.11} \end{equation*}(3.11)Qδc3ω1≅C[a0,a1,a2]/(a02,a0a1,a0a2+a12)
Both (3.10) and (3.11) follow from Conjecture 3.12.4, and both can be proved by direct computation in G r μ G r μ Gr^(mu)\mathrm{Gr}^{\mu}Grμ as explained above.
Remark 3.15. It is surprising how complex J d ( Spec ( H ( Gr ( k , n ) , C ) ) ) J d Spec ⁡ H ∗ ( Gr ⁡ ( k , n ) , C ) J_(d)(Spec(H^(**)(Gr(k,n),C)))J_{d}\left(\operatorname{Spec}\left(H^{*}(\operatorname{Gr}(k, n), \mathbb{C})\right)\right)Jd(Spec⁡(H∗(Gr⁡(k,n),C))) can be. In particular, in the k = 1 k = 1 k=1k=1k=1 case (i.e., jet schemes of the cohomology ring of projective space) there is only a conjecture about its multiplicity in [50, CONJECTURE III.21].
Remark 3.16. Finally, we remark that already for type (2) we have new phenomena. As discussed in [ 37 , $ 5.4 ] [ 37 , $ 5.4 ] [37,$5.4][37, \$ 5.4][37,$5.4], there are multiplicity algebras depending on continuous parameters, in particular they cannot be isomorphic to cohomology rings, because cohomology rings are integral.

3.3.1. Lagrangian closure of W δ + W δ + W_(delta)^(+)W_{\delta}^{+}Wδ+

Definition 3.17. Let E M s π E ∈ M s Ï€ EinM^(s pi)\mathscr{E} \in \mathcal{M}^{s \pi}E∈MsÏ€. The Lagrangian closure W E + ¯ ¯ W E + ¯ ¯ bar(bar(W_(E)^(+)))\overline{\overline{W_{\mathcal{E}}^{+}}}WE+¯¯of W E + W E + W_(E)^(+)W_{\mathcal{E}}^{+}WE+is the smallest closed union of upward flows containing W E + W E + W_(E)^(+)W_{\mathcal{E}}^{+}WE+. In other words, the Lagrangian closure is the closure in the quotient space by the BB partition.
Using (3.8) and (3.4), we can deduce the following
Theorem 3.18 ([30]). Let μ P + μ ∈ P + mu inP^(+)\mu \in P^{+}μ∈P+and c C c ∈ C c in Cc \in Cc∈C. Recall δ c μ δ c μ delta_(c)^(mu)\delta_{c}^{\mu}δcμ from (3.7). Assume E δ c μ M s π E δ c μ ∈ M s Ï€ E_(delta_(c)^(mu))inM^(s pi)E_{\delta_{c}^{\mu}} \in \mathcal{M}^{s \pi}Eδcμ∈MsÏ€. Then
W E + ¯ ¯ = μ λ P + W δ c λ + W E + ¯ ¯ = ∐ μ ≥ λ ∈ P +   W δ c λ + bar(bar(W_(E)^(+)))=∐_(mu >= lambda inP^(+))W_(delta_(c)^(lambda))^(+)\overline{\overline{W_{\mathcal{E}}^{+}}}=\coprod_{\mu \geq \lambda \in P^{+}} W_{\delta_{c}^{\lambda}}^{+}WE+¯¯=∐μ≥λ∈P+Wδcλ+
i.e., the upward flows correspond to dominant weights λ λ lambda\lambdaλ less than or equal to μ μ mu\muμ in dominance order.

3.4. Towards a classical limit of the geometric Satake correspondence

Finally, we will formulate some conjectures which were the original motivation of much of the previous ideas. In particular, they hint at a new construction of the irreducible representations of G L n ( C ) G L n ( C ) GL_(n)(C)\mathrm{GL}_{n}(\mathbb{C})GLn(C), and more generally of any complex reductive group G G G\mathrm{G}G.
The general setup comes from the classical limit (2.5) of the geometric Langlands program, as formulated in [13]. Here we sketch some of the expectations of this classical limit in a schematic (not completely well defined) manner. It should be an equivalence of some sort of derived categories of coherent sheaves
S : D b ( M D o l ) D b ( M D o l ) S : D b M D o l → D b M D o l ∨ S:D^(b)(M_(Dol))rarrD^(b)(M_(Dol)^(vv))\mathscr{S}: D^{b}\left(\mathcal{M}_{\mathrm{Dol}}\right) \rightarrow D^{b}\left(\mathcal{M}_{\mathrm{Dol}}^{\vee}\right)S:Db(MDol)→Db(MDol∨)
Several properties of this equivalence were proposed and some established in [13]. In particular, S S S\mathscr{S}S should be a relative Fourier-Mukai transform along the generic locus in A G A G A G ≅ A G A_(G)~=A_(G)\mathcal{A}_{\mathrm{G}} \cong \mathscr{A}_{\mathrm{G}}AG≅AG. Another crucial property [38], which we called enhanced mirror symmetry in Section 2.5 above, is that S S S\mathscr{S}S should intertwine the actions of certain Hecke operators on D b ( M Dol ) D b M Dol  D^(b)(M_("Dol "))D^{b}\left(\mathcal{M}_{\text {Dol }}\right)Db(MDol ) and the Wilson operators on D b ( M Dol ) D b M Dol  D^(b)(M_("Dol "))D^{b}\left(\mathcal{M}_{\text {Dol }}\right)Db(MDol ). Let μ X + ( G ) = X + ( G ) μ ∈ X + G ∨ = X + ( G ) mu inX_(+)(G^(vv))=X^(+)(G)\mu \in X_{+}\left(\mathrm{G}^{\vee}\right)=X^{+}(\mathrm{G})μ∈X+(G∨)=X+(G) be a dominant character of G G ∨ G^(vv)\mathrm{G}^{\vee}G∨. We denote by
H μ := { ( E , Φ ) M D o l , [ g ] G r μ g 1 Φ c g G [ [ z ] ] } M D o l × G r μ H μ := ( E , Φ ) ∈ M D o l , [ g ] ∈ G r μ ∣ g − 1 Φ c g ∈ G [ [ z ] ] ⊂ M D o l × G r μ H^(mu):={(E,Phi)inM_(Dol),[g]inGr^(mu)∣g^(-1)Phi_(c)g inG[[z]]}subM_(Dol)xxGr^(mu)\mathscr{H}^{\mu}:=\left\{(E, \Phi) \in \mathcal{M}_{\mathrm{Dol}},[g] \in \mathrm{Gr}^{\mu} \mid g^{-1} \Phi_{c} g \in \mathrm{G}[[z]]\right\} \subset \mathcal{M}_{\mathrm{Dol}} \times \mathrm{Gr}^{\mu}Hμ:={(E,Φ)∈MDol,[g]∈Grμ∣g−1Φcg∈G[[z]]}⊂MDol×Grμ
some space of Hecke correspondences at a point c C c ∈ C c in Cc \in Cc∈C. Indeed, this gives us
two maps to M Dol M Dol  M_("Dol ")\mathcal{M}_{\text {Dol }}MDol , first the projection π μ Ï€ μ pi_(mu)\pi_{\mu}πμ to the first factor, and second f μ f μ f^(mu)f^{\mu}fμ, the Hecke transformation 1 1 ^(1){ }^{1}1 of ( E , Φ ) ( E , Φ ) (E,Phi)(E, \Phi)(E,Φ) by the compatible Hecke transform [ g ] G r μ [ g ] ∈ G r μ [g]inGr^(mu)[g] \in \mathrm{Gr}^{\mu}[g]∈Grμ, which is expected to induce
H μ := f μ π μ : D b ( M D o l ) D b ( M D o l ) H μ := f ∗ μ ∘ Ï€ μ ∗ : D b M D o l → D b M D o l H^(mu):=f_(**)^(mu)@pi_(mu)^(**):D^(b)(M_(Dol))rarrD^(b)(M_(Dol))\mathscr{H}^{\mu}:=f_{*}^{\mu} \circ \pi_{\mu}^{*}: D^{b}\left(\mathcal{M}_{\mathrm{Dol}}\right) \rightarrow D^{b}\left(\mathcal{M}_{\mathrm{Dol}}\right)Hμ:=f∗μ∘πμ∗:Db(MDol)→Db(MDol)
a Hecke (or the physicists' t'Hooft) operator.
On the other hand, we can consider the so-called Wilson operators
W μ : D b ( M D o l ) D b ( M D o l ) , F F ρ μ ( E | M D o l × { c } ) W μ : D b M D o l ∨ → D b M D o l ∨ , F ↦ F ⊗ ρ μ E M D o l ∨ × { c } {:[W^(mu):D^(b)(M_(Dol)^(vv)) rarrD^(b)(M_(Dol)^(vv))","],[F|->Foxrho_(mu)(E|_(M_(Dol)^(vv)xx{c}))]:}\begin{aligned} \mathscr{W}^{\mu}: D^{b}\left(\mathcal{M}_{\mathrm{Dol}}^{\vee}\right) & \rightarrow D^{b}\left(\mathcal{M}_{\mathrm{Dol}}^{\vee}\right), \\ \mathscr{F} & \mapsto \mathscr{F} \otimes \rho_{\mu}\left(\left.\mathbb{E}\right|_{\mathcal{M}_{\mathrm{Dol}}^{\vee} \times\{c\}}\right) \end{aligned}Wμ:Db(MDol∨)→Db(MDol∨),F↦F⊗ρμ(E|MDol∨×{c})
given by tensoring with the universal G G ∨ G^(vv)\mathrm{G}^{\vee}G∨ bundle E E E\mathbb{E}E in the representation ρ μ : G G L ( V ρ μ ) ρ μ : G ∨ → G L V ρ μ rho_(mu):G^(vv)rarrGL(V_(rho_(mu)))\rho_{\mu}: \mathrm{G}^{\vee} \rightarrow \mathrm{GL}\left(V_{\rho_{\mu}}\right)ρμ:G∨→GL(Vρμ).
1 Here we ignore stability issues.
We then expect [ 13 , 38 ] [ 13 , 38 ] [13,38][13,38][13,38] that
(3.12) W μ S = S H μ (3.12) W μ ∘ S = S ∘ H μ {:(3.12)W^(mu)@S=S@H^(mu):}\begin{equation*} \mathscr{W}^{\mu} \circ \mathscr{S}=\mathscr{S} \circ \mathscr{H}^{\mu} \tag{3.12} \end{equation*}(3.12)Wμ∘S=S∘Hμ
There are two more expectations for the classical limit S S S\mathscr{S}S, both are motivated from Fourier-Mukai transform where the analogous statements hold. First, we expect that for any F D b ( M Dol ) F ∈ D b M Dol  FinD^(b)(M_("Dol "))\mathscr{F} \in D^{b}\left(\mathcal{M}_{\text {Dol }}\right)F∈Db(MDol ) we should have
(3.13) ( h G ) ( F ) S ( F ) | W 0 + (3.13) h G ∗ ( F ) ≅ S ( F ) W 0 + {:(3.13)(h_(G))_(**)(F)~=S(F)|_(W_(0)^(+)):}\begin{equation*} \left.\left(h_{\mathrm{G}}\right)_{*}(\mathcal{F}) \cong \mathscr{S}(\mathscr{F})\right|_{W_{0}^{+}} \tag{3.13} \end{equation*}(3.13)(hG)∗(F)≅S(F)|W0+
Second, the structure sheaf of the Hitchin section should be mirror to the structure sheaf of the mirror Higgs moduli space,
(3.14) S ( O W 0 + ) O M D o l (3.14) S O W 0 + ≅ O M D o l ∨ {:(3.14)S(O_(W_(0)^(+)))~=O_(M_(Dol)^(vv)):}\begin{equation*} \mathscr{S}\left(\mathcal{O}_{W_{0}^{+}}\right) \cong \mathcal{O}_{\mathcal{M}_{\mathrm{Dol}}^{\vee}} \tag{3.14} \end{equation*}(3.14)S(OW0+)≅OMDol∨
Combining (3.12) with (3.14), we can deduce that
S ( H μ ( O W 0 + ) ) = W μ ( S ( O W 0 + ) ) = W μ ( O M D o l ) S H μ O W 0 + = W μ S O W 0 + = W μ O M D o l ∨ S(H^(mu)(O_(W_(0)^(+))))=W^(mu)(S(O_(W_(0)^(+))))=W^(mu)(O_(M_(Dol)^(vv)))\mathscr{S}\left(\mathscr{H}^{\mu}\left(\mathcal{O}_{W_{0}^{+}}\right)\right)=\mathscr{W}^{\mu}\left(\mathscr{S}\left(\mathcal{O}_{W_{0}^{+}}\right)\right)=\mathscr{W}^{\mu}\left(\mathcal{O}_{\mathcal{M}_{\mathrm{Dol}}^{\vee}}\right)S(Hμ(OW0+))=Wμ(S(OW0+))=Wμ(OMDol∨)
On the one hand, we should have
supp ( H μ ( O W 0 + ) ) = W μ + ¯ ¯ supp ⁡ H μ O W 0 + = W μ + ¯ ¯ supp(H^(mu)(O_(W_(0)^(+))))= bar(bar(W_(mu)^(+)))\operatorname{supp}\left(\mathscr{H}^{\mu}\left(\mathcal{O}_{W_{0}^{+}}\right)\right)=\overline{\overline{W_{\mu}^{+}}}supp⁡(Hμ(OW0+))=Wμ+¯¯
where W μ + W μ + W_(mu)^(+)W_{\mu}^{+}Wμ+is the upward flow from a certain ε μ ε μ epsi_(mu)\varepsilon_{\mu}εμ maximally split G G GGG-Higgs bundle of type μ μ mu\muμ at c C c ∈ C c in Cc \in Cc∈C. On the other hand,
W μ ( O M D o l ) = ρ μ ( E ) c =: Λ μ W μ O M D o l ∨ ∨ = ρ μ ( E ) c =: Λ μ W^(mu)(O_(M_(Dol)^(vv))^(vv))=rho_(mu)(E)_(c)=:Lambda_(mu)\mathscr{W}^{\mu}\left(\mathcal{O}_{\mathcal{M}_{\mathrm{Dol}}^{\vee}}^{\vee}\right)=\rho_{\mu}(\mathbb{E})_{c}=: \Lambda_{\mu}Wμ(OMDol∨∨)=ρμ(E)c=:Λμ
the vector bundle associated to the principal bundle E c E c E_(c)\mathbb{E}_{c}Ec in the representation ρ μ ρ μ rho_(mu)\rho_{\mu}ρμ.
Thus Kapustin-Witten's (3.12) implies
S ( H μ ( O W 0 + ) ) = Λ μ S H μ O W 0 + = Λ μ S(H^(mu)(O_(W_(0)^(+))))=Lambda_(mu)\mathscr{S}\left(\mathscr{H}^{\mu}\left(\mathcal{O}_{W_{0}^{+}}\right)\right)=\Lambda_{\mu}S(Hμ(OW0+))=Λμ
We can test this by (3.13) as
Λ μ | L W 0 + = S ( H μ ( O W 0 + ) ) | L W 0 + = ( h G ) ( H μ ( O W 0 + ) ) Λ μ L W 0 + = S H μ O W 0 + L W 0 + = h G ∗ H μ O W 0 + Lambda_(mu)|_(LW_(0)^(+))=S(H^(mu)(O_(W_(0)^(+))))|_(L_(W_(0)^(+)))=(h_(G))_(**)(H^(mu)(O_(W_(0)^(+))))\left.\Lambda_{\mu}\right|_{L W_{0}^{+}}=\left.\mathscr{S}\left(\mathscr{H}^{\mu}\left(\mathcal{O}_{W_{0}^{+}}\right)\right)\right|_{L_{W_{0}^{+}}}=\left(h_{\mathrm{G}}\right)_{*}\left(\mathscr{H}^{\mu}\left(\mathcal{O}_{W_{0}^{+}}\right)\right)Λμ|LW0+=S(Hμ(OW0+))|LW0+=(hG)∗(Hμ(OW0+))
In [29] we have argued that the mirror of the structure sheaf of a very stable type ( 1 , , 1 ) ( 1 , … , 1 ) (1,dots,1)(1, \ldots, 1)(1,…,1) upward flow W δ W δ W_(delta)W_{\delta}Wδ is
Λ δ := i = 1 n 1 j = 1 m i Λ i E c i j Λ δ := ⨂ i = 1 n − 1   ⨂ j = 1 m i   Λ i E c i j Lambda_(delta):=⨂_(i=1)^(n-1)⨂_(j=1)^(m_(i))Lambda^(i)E_(c_(ij))\Lambda_{\delta}:=\bigotimes_{i=1}^{n-1} \bigotimes_{j=1}^{m_{i}} \Lambda^{i} \mathbb{E}_{c_{i j}}Λδ:=⨂i=1n−1⨂j=1miΛiEcij
where ( E , d ) ( E , d ) (E,d)(\mathbb{E}, d)(E,d) is a universal Higgs bundle on M × C M × C Mxx C\mathcal{M} \times CM×C and
δ i = c i 1 + c i 2 + C [ m i ] δ i = c i 1 + c i 2 + ⋯ ∈ C m i delta_(i)=c_(i1)+c_(i2)+cdots inC^([m_(i)])\delta_{i}=c_{i 1}+c_{i 2}+\cdots \in C^{\left[m_{i}\right]}δi=ci1+ci2+⋯∈C[mi]
In particular, one expectation of mirror symmetry is that
h ( O W δ + ) Λ δ | W 0 + h ∗ O W δ + ≅ Λ δ W 0 + h_(**)(O_(W_(delta)^(+)))~=Lambda_(delta)|_(W_(0)^(+))\left.h_{*}\left(\mathcal{O}_{W_{\delta}^{+}}\right) \cong \Lambda_{\delta}\right|_{W_{0}^{+}}h∗(OWδ+)≅Λδ|W0+
This follows from Theorem 3.6 and a direct computation for χ T ( E c ) χ T E c chi_(T)(E_(c))\chi_{\mathbb{T}}\left(\mathbb{E}_{c}\right)χT(Ec).
In [ 29 , $ 8.2 ] [ 29 , $ 8.2 ] [29,$8.2][29, \$ 8.2][29,$8.2] we proposed that for n = 2 n = 2 n=2n=2n=2 the mirror of Sym 2 ( E c ) Sym 2 ⁡ E c Sym^(2)(E_(c))\operatorname{Sym}^{2}\left(\mathbb{E}_{c}\right)Sym2⁡(Ec) should be the struc-
ture sheaf of the Lagrangian closure W δ c 2 + ¯ ¯ W δ c 2 + ¯ ¯ bar(bar(W_(delta_(c)^(2))^(+)))\overline{\overline{W_{\delta_{c}^{2}}^{+}}}Wδc2+¯¯where δ c 2 = ( 0 , 2 c ) δ c 2 = ( 0 , 2 c ) delta_(c)^(2)=(0,2c)\delta_{c}^{2}=(0,2 c)δc2=(0,2c). We can generalize this as follows.
Conjecture 3.19. Let c C c ∈ C c in Cc \in Cc∈C and G G G\mathrm{G}G a reductive group. Then we have the following conjectures:
(1) For any μ X + ( G ) μ ∈ X + G ∨ mu inX_(+)(G^(vv))\mu \in X_{+}\left(\mathrm{G}^{\vee}\right)μ∈X+(G∨), the support of the mirror of ρ μ ( E c ) ρ μ E c rho_(mu)(E_(c))\rho_{\mu}\left(\mathbb{E}_{c}\right)ρμ(Ec) is W δ c μ + ¯ ¯ W δ c μ + ¯ ¯ bar(bar(W_(delta_(c)^(mu))^(+)))\overline{\overline{W_{\delta_{c}^{\mu}}^{+}}}Wδcμ+¯¯.
(2) Let μ X + ( G ) μ ∈ X + G ∨ mu inX_(+)(G^(vv))\mu \in X_{+}\left(\mathrm{G}^{\vee}\right)μ∈X+(G∨) such that the corresponding irreducible G G ∨ G^(vv)\mathrm{G}^{\vee}G∨ representation ρ μ ρ μ rho_(mu)\rho_{\mu}ρμ is multiplicity free. Then the mirror of ρ μ ( E c ) ρ μ E c rho_(mu)(E_(c))\rho_{\mu}\left(\mathbb{E}_{c}\right)ρμ(Ec) is O W δ c + + ¯ ¯ O W δ c + + ¯ ¯ O bar(bar(W_(delta_(c)^(+))^(+)))\mathcal{O} \overline{\overline{W_{\delta_{c}^{+}}^{+}}}OWδc++¯¯.
(3) In the latter case, the multiplicity algebra of the restricted Hitchin map h G h G h_(G)h_{\mathrm{G}}hG : W δ c μ + ¯ ¯ A W δ c μ + ¯ ¯ → A bar(bar(W_(delta_(c)^(mu))^(+)))rarrA\overline{\overline{W_{\delta_{c}^{\mu}}^{+}}} \rightarrow \mathcal{A}Wδcμ+¯¯→A is isomorphic with the cohomology ring of G r ¯ μ G r ¯ μ bar(Gr)^(mu)\overline{\mathrm{Gr}}^{\mu}Gr¯μ.
Remark 3.20. In [17], studying opers in the geometric Langlands program, the authors construct a canonical Poincaré duality ring structure on the underlying vector space V μ V μ V_(mu)V_{\mu}Vμ of all irreducible representation ρ μ ρ μ rho_(mu)\rho_{\mu}ρμ of G G ∨ G^(vv)\mathrm{G}^{\vee}G∨. In the case when ρ μ ρ μ rho_(mu)\rho_{\mu}ρμ is multiplicity-free, this ring is isomorphic with the cohomology ring H ( G r ¯ μ ) H ∗ G r ¯ μ H^(**)( bar(Gr)^(mu))H^{*}\left(\overline{\mathrm{Gr}}^{\mu}\right)H∗(Gr¯μ). Note that, according to [20, THEOREM 1.5], these are precisely the cases when
H ( G r ¯ μ ) I H ( G r ¯ μ ) H ∗ G r ¯ μ ≅ I H ∗ G r ¯ μ H^(**)( bar(Gr)^(mu))~=IH^(**)( bar(Gr)^(mu))H^{*}\left(\overline{\mathrm{Gr}}^{\mu}\right) \cong I H^{*}\left(\overline{\mathrm{Gr}}^{\mu}\right)H∗(Gr¯μ)≅IH∗(Gr¯μ)
when the cohomology ring satisfies Poincaré duality. In this case, this ring was more carefully studied in [47].

ACKNOWLEDGMENTS

The author thanks Nigel Hitchin for introducing him to Higgs bundles during 1995-1998, suggesting the SYZ picture for Langlands dual Hitchin systems in 1996, and for the more recent collaborations [29, 30]. He also thanks David Ben-Zvi, Pierre-Henri Chaudouard, Pierre Deligne, Ron Donagi, Sergei Gukov, Jochen Heinloth, Vadim Kaloshin, Joel Kamnitzer, Gérard Laumon, Anton Mellit, David Nadler, Andy Neitzke, Ngô Bao Châu, Michael Thaddeus, Tony Pantev, Du Pei, Richárd Rimányi, Leonid Rybnikov, Vivek Shende, Balázs Szendrő́i, András Szenes, Fernando Rodriguez-Villegas, Edward Witten, and Zhiwei Yun for discussions about the subjects in this paper over the years. Thanks are also due to Hülya Argüz, Jakub Löwit, Balázs Szendrői, and Nigel Hitchin for the careful reading of the paper.

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TAMÁS HAUSEL

Institute of Science and Technology Austria, Am Campus 1, Klosterneuburg 3400, Austria, tamas.hausel@ist.ac.at

HODGE THEORY, BETWEEN ALGEBRAICITY AND TRANSCENDENCE
BRUNO KLINGLER

ABSTRACT

The Hodge theory of complex algebraic varieties is at heart a transcendental comparison of two algebraic structures. We survey the recent advances bounding this transcendence, mainly due to the introduction of o-minimal geometry as a natural framework for Hodge theory.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 14D; Secondary 14D07, 14C30, 32G20

KEYWORDS

Hodge theory, variations of Hodge structures, periods

1. INTRODUCTION

Let X X XXX be a smooth connected projective variety over C C C\mathbb{C}C, and X an X an  X^("an ")X^{\text {an }}Xan  its associated compact complex manifold. Classical Hodge theory [52] states that the Betti (i.e., singular) cohomology group H B k ( X an , Z ) H B k X an  , Z H_(B)^(k)(X^("an "),Z)H_{\mathrm{B}}^{k}\left(X^{\text {an }}, \mathbb{Z}\right)HBk(Xan ,Z) is a polarizable Z Z Z\mathbb{Z}Z-Hodge structure of weight k k kkk : there exists a canonical decomposition (called the Hodge decomposition) of complex vector spaces
H B k ( X a n , Z ) Z C = p + q = k H p , q ( X a n ) satisfying H p , q ( X a n ) ¯ = H q , p ( X a n ) H B k X a n , Z ⊗ Z C = ⨁ p + q = k   H p , q X a n  satisfying  H p , q X a n ¯ = H q , p X a n H_(B)^(k)(X^(an),Z)ox_(Z)C=bigoplus_(p+q=k)H^(p,q)(X^(an))quad" satisfying " bar(H^(p,q)(X^(an)))=H^(q,p)(X^(an))H_{\mathrm{B}}^{k}\left(X^{\mathrm{an}}, \mathbb{Z}\right) \otimes_{\mathbb{Z}} \mathbb{C}=\bigoplus_{p+q=k} H^{p, q}\left(X^{\mathrm{an}}\right) \quad \text { satisfying } \overline{H^{p, q}\left(X^{\mathrm{an}}\right)}=H^{q, p}\left(X^{\mathrm{an}}\right)HBk(Xan,Z)⊗ZC=⨁p+q=kHp,q(Xan) satisfying Hp,q(Xan)¯=Hq,p(Xan)
and a ( 1 ) k ( − 1 ) k (-1)^(k)(-1)^{k}(−1)k-symmetric bilinear pairing q k : H B k ( X an , Z ) × H B k ( X an , Z ) Z q k : H B k X an  , Z × H B k X an  , Z → Z q_(k):H_(B)^(k)(X^("an "),Z)xxH_(B)^(k)(X^("an "),Z)rarrZq_{k}: H_{\mathrm{B}}^{k}\left(X^{\text {an }}, \mathbb{Z}\right) \times H_{\mathrm{B}}^{k}\left(X^{\text {an }}, \mathbb{Z}\right) \rightarrow \mathbb{Z}qk:HBk(Xan ,Z)×HBk(Xan ,Z)→Z whose complexification makes the above decomposition orthogonal, and satisfies the positivity condition (the signs are complicated but are imposed to us by geometry)
i p q q k , C ( α , α ¯ ) > 0 for any nonzero α H p , q ( X a n ) i p − q q k , C ( α , α ¯ ) > 0  for any nonzero  α ∈ H p , q X a n i^(p-q)q_(k,C)(alpha, bar(alpha)) > 0quad" for any nonzero "alpha inH^(p,q)(X^(an))\mathrm{i}^{p-q} q_{k, \mathbb{C}}(\alpha, \bar{\alpha})>0 \quad \text { for any nonzero } \alpha \in H^{p, q}\left(X^{\mathrm{an}}\right)ip−qqk,C(α,α¯)>0 for any nonzero α∈Hp,q(Xan)
Deligne [29] vastly generalized Hodge's result, showing that the cohomology H B k ( X an , Z ) H B k X an  , Z H_(B)^(k)(X^("an "),Z)H_{\mathrm{B}}^{k}\left(X^{\text {an }}, \mathbb{Z}\right)HBk(Xan ,Z) of any complex algebraic variety X X XXX is functorially endowed with a slightly more general graded polarizable mixed Z Z Z\mathbb{Z}Z-Hodge structure, that makes, after tensoring with Q , H B k ( X a n , Q ) Q , H B k X a n , Q Q,H_(B)^(k)(X^(an),Q)\mathbb{Q}, H_{\mathrm{B}}^{k}\left(X^{\mathrm{an}}, \mathbb{Q}\right)Q,HBk(Xan,Q) a successive extension of polarizable Q Q Q\mathbb{Q}Q-Hodge structures, with weights between 0 and 2 k 2 k 2k2 k2k. As mixed Q Q Q\mathbb{Q}Q-Hodge structures form a Tannakian category M H S Q M H S Q MHS_(Q)\mathrm{MHS}_{\mathbb{Q}}MHSQ, one can conveniently (although rather abstractly) summarize the Hodge-Deligne theory as functorially assigning to any complex algebraic variety X X XXX a Q Q Q\mathbb{Q}Q-algebraic group: the Mumford-Tate group M T X M T X MT_(X)\mathbf{M T}_{X}MTX of X X XXX, defined as the Tannaka group of the Tannakian subcategory H B ( X an , Q ) H B ∙ X an  , Q (:H_(B)^(∙)(X^("an "),Q):)\left\langle H_{\mathrm{B}}^{\bullet}\left(X^{\text {an }}, \mathbb{Q}\right)\right\rangle⟨HB∙(Xan ,Q)⟩ of M H S Q M H S Q MHS_(Q)\mathrm{MHS}_{\mathbb{Q}}MHSQ generated by H B ( X an , Q ) H B ∙ X an  , Q H_(B)^(∙)(X^("an "),Q)H_{\mathrm{B}}^{\bullet}\left(X^{\text {an }}, \mathbb{Q}\right)HB∙(Xan ,Q). The knowledge of the group M T X M T X MT_(X)\mathbf{M} \mathbf{T}_{X}MTX is equivalent to the knowledge of all Hodge tensors for the Hodge structure H B ( X an , Q ) H B ∙ X an  , Q H_(B)^(∙)(X^("an "),Q)H_{\mathrm{B}}^{\bullet}\left(X^{\text {an }}, \mathbb{Q}\right)HB∙(Xan ,Q).
These apparently rather innocuous semilinear algebra statements are anything but trivial. They have become the main tool for analyzing the topology, geometry and arithmetic of complex algebraic varieties. Let us illustrate what we mean with regard to topology, which we will not go into later. The existence of the Hodge decomposition for smooth projective complex varieties, which holds more generally for compact Kähler manifolds, imposes many constraints on the cohomology of such spaces, the most obvious being that their odd Betti numbers have to be even. Such constraints are not satisfied even by compact complex manifolds as simple as the Hopf surfaces, quotients of C 2 { 0 } C 2 ∖ { 0 } C^(2)\\{0}\mathbb{C}^{2} \backslash\{0\}C2∖{0} by the action of Z Z Z\mathbb{Z}Z given by multiplication by λ 0 , | λ | 1 λ ≠ 0 , | λ | ≠ 1 lambda!=0,|lambda|!=1\lambda \neq 0,|\lambda| \neq 1λ≠0,|λ|≠1, whose first Betti number is one. Characterizing the homotopy types of compact Kähler manifolds is an essentially open question, which we will not discuss here.
The mystery of the Hodge-Deligne theory lies in the fact that it is at heart not an algebraic theory, but rather the transcendental comparison of two algebraic structures. For simplicity, let X X XXX be a smooth connected projective variety over C C C\mathbb{C}C. The Betti cohomology H B ( X an , Q ) H B ∙ X an  , Q H_(B)^(∙)(X^("an "),Q)H_{\mathrm{B}}^{\bullet}\left(X^{\text {an }}, \mathbb{Q}\right)HB∙(Xan ,Q) defines a Q Q Q\mathbb{Q}Q-structure on the complex vector space of the algebraic de Rham cohomology H d R ( X / C ) := H ( X , Ω X / C ) H d R ∙ ( X / C ) := H ∙ X , Ω X / C ∙ H_(dR)^(∙)(X//C):=H^(∙)(X,Omega_(X//C)^(∙))H_{\mathrm{dR}}^{\bullet}(X / \mathbb{C}):=H^{\bullet}\left(X, \Omega_{X / \mathbb{C}}^{\bullet}\right)HdR∙(X/C):=H∙(X,ΩX/C∙) under the transcendental comparison isomorphism:
(1.1) ϖ : H d R ( X / C ) H ( X a n , Ω X ) =: H d R ( X a n , C ) H B ( X a n , Q ) Q C (1.1) Ï– : H d R ∙ ( X / C ) → ∼ H ∙ X a n , Ω X ∙ =: H d R ∙ X a n , C → ∼ H B ∙ X a n , Q ⊗ Q C {:(1.1)Ï–:H_(dR)^(∙)(X//C)rarr"∼"H^(∙)(X^(an),Omega_(X^(∙)))=:H_(dR)^(∙)(X^(an),C)rarr"∼"H_(B)^(∙)(X^(an),Q)ox_(Q)C:}\begin{equation*} \varpi: H_{\mathrm{dR}}^{\bullet}(X / \mathbb{C}) \xrightarrow{\sim} H^{\bullet}\left(X^{\mathrm{an}}, \Omega_{X^{\bullet}}\right)=: H_{\mathrm{dR}}^{\bullet}\left(X^{\mathrm{an}}, \mathbb{C}\right) \xrightarrow{\sim} H_{\mathrm{B}}^{\bullet}\left(X^{\mathrm{an}}, \mathbb{Q}\right) \otimes_{\mathbb{Q}} \mathbb{C} \tag{1.1} \end{equation*}(1.1)Ï–:HdR∙(X/C)→∼H∙(Xan,ΩX∙)=:HdR∙(Xan,C)→∼HB∙(Xan,Q)⊗QC
where the first canonical isomorphism is the comparison between algebraic and analytic de Rham cohomology provided by GAGA, and the second one is provided by integrating complex C C ∞ C^(oo)\mathrm{C}^{\infty}C∞ differential forms over cycles (de Rham's theorem). The Hodge filtration F p F p F^(p)F^{p}Fp on H B ( X an , Q ) Q C H B ∙ X an  , Q ⊗ Q C H_(B)^(∙)(X^("an "),Q)ox_(Q)CH_{\mathrm{B}}^{\bullet}\left(X^{\text {an }}, \mathbb{Q}\right) \otimes_{\mathbb{Q}} \mathbb{C}HB∙(Xan ,Q)⊗QC is the image under ϖ Ï– Ï–\varpiÏ– of the algebraic filtration F p = F p = F^(p)=F^{p}=Fp= Im ( H ( X , Ω X / C p ) H d R ( X / C ) ) Im ⁡ H ∙ X , Ω X / C ∙ ≥ p → H d R ∙ ( X / C ) Im(H^(∙)(X,Omega_(X//C)^(∙ >= p))rarrH_(dR)^(∙)(X//C))\operatorname{Im}\left(H^{\bullet}\left(X, \Omega_{X / \mathbb{C}}^{\bullet \geq p}\right) \rightarrow H_{\mathrm{dR}}^{\bullet}(X / \mathbb{C})\right)Im⁡(H∙(X,ΩX/C∙≥p)→HdR∙(X/C)) on the left-hand side.
The surprising power of the Hodge-Deligne theory lies in the fact that, although the comparison between the two algebraic structures is transcendental, this transcendence should be severely constrained, as predicted, for instance, by the Hodge conjecture and the Grothendieck period conjecture:
  • For X X XXX smooth projective, it is well known that the cycle class [ Z ] [ Z ] [Z][Z][Z] of any codimension k k kkk algebraic cycle on X X XXX with Q Q Q\mathbb{Q}Q coefficients is a Hodge class in the Hodge structure H 2 k ( X an , Q ) ( k ) H 2 k X an  , Q ( k ) H^(2k)(X^("an "),Q)(k)H^{2 k}\left(X^{\text {an }}, \mathbb{Q}\right)(k)H2k(Xan ,Q)(k). Hodge [52] famously conjectured that the converse holds true: any Hodge class in H 2 k ( X , Q ) ( k ) H 2 k ( X , Q ) ( k ) H^(2k)(X,Q)(k)H^{2 k}(X, \mathbb{Q})(k)H2k(X,Q)(k) should be such a cycle class.
  • For X X XXX smooth and defined over a number field K C K ⊂ C K subCK \subset \mathbb{C}K⊂C, its periods are the coefficients of the matrix of Grothendieck's isomorphism (generalizing (1.1))
ϖ : H d R ( X / K ) K C H B ( X a n , Q ) Q C Ï– : H d R ∙ ( X / K ) ⊗ K C → ∼ H B ∙ X a n , Q ⊗ Q C Ï–:H_(dR)^(∙)(X//K)ox_(K)Crarr"∼"H_(B)^(∙)(X^(an),Q)ox_(Q)C\varpi: H_{\mathrm{dR}}^{\bullet}(X / K) \otimes_{K} \mathbb{C} \xrightarrow{\sim} H_{\mathrm{B}}^{\bullet}\left(X^{\mathrm{an}}, \mathbb{Q}\right) \otimes_{\mathbb{Q}} \mathbb{C}Ï–:HdR∙(X/K)⊗KC→∼HB∙(Xan,Q)⊗QC
with respect to bases of H d R ( X / K ) H d R ∙ ( X / K ) H_(dR)^(∙)(X//K)H_{\mathrm{dR}}^{\bullet}(X / K)HdR∙(X/K) and H B ( X an , Q ) H B ∙ X an  , Q H_(B)^(∙)(X^("an "),Q)H_{\mathrm{B}}^{\bullet}\left(X^{\text {an }}, \mathbb{Q}\right)HB∙(Xan ,Q). The Grothendieck period conjecture (combined with the Hodge conjecture) predicts that the transcendence degree of the field k X C k X ⊂ C k_(X)subCk_{X} \subset \mathbb{C}kX⊂C generated by the periods of X X XXX coincides with the dimension of M T X M T X MT_(X)\mathbf{M} \mathbf{T}_{X}MTX.
This tension between algebraicity and transcendence is perhaps best revealed when considering Hodge theory in families, as developed by Griffiths [43]. Let f : X S f : X → S f:X rarr Sf: X \rightarrow Sf:X→S be a smooth projective morphism of smooth connected quasiprojective varieties over C C C\mathbb{C}C. Its complex analytic fibers X s a n , s S a n X s a n , s ∈ S a n X_(s)^(an),s inS^(an)X_{s}^{\mathrm{an}}, s \in S^{\mathrm{an}}Xsan,s∈San, are diffeomorphic, hence their cohomologies V Z , s := H B ( X s a n , Z ) V Z , s := H B ∙ X s a n , Z V_(Z,s):=H_(B)^(∙)(X_(s)^(an),Z)\mathbb{V}_{\mathbb{Z}, s}:=H_{\mathrm{B}}^{\bullet}\left(X_{s}^{\mathrm{an}}, \mathbb{Z}\right)VZ,s:=HB∙(Xsan,Z), s S an s ∈ S an  s inS^("an ")s \in S^{\text {an }}s∈San  are all isomorphic to a fixed abelian group V Z V Z V_(Z)V_{\mathbb{Z}}VZ and glue together into a locally constant sheaf V Z := R f an V Z := R ∙ f an  ∗ V_(Z):=R^(∙)f^("an ")_(**)\mathbb{V}_{\mathbb{Z}}:=R^{\bullet} f^{\text {an }}{ }_{*}VZ:=R∙fan ∗ on S an S an  S^("an ")S^{\text {an }}San . However, the complex algebraic structure on X s X s X_(s)X_{s}Xs, hence also the Hodge structure on V Z , s V Z , s V_(Z,s)\mathbb{V}_{\mathbb{Z}, s}VZ,s, varies with s s sss, making R f an Z R ∙ f ∗ an  Z R^(∙)f_(**)^("an ")ZR^{\bullet} f_{*}^{\text {an }} \mathbb{Z}R∙f∗an Z a variation of Z Z Z\mathbb{Z}Z-Hodge structures ( Z V H S ) V Z V H S ) V ZVHS)V\mathbb{Z V H S}) \mathbb{V}ZVHS)V on S an S an  S^("an ")S^{\text {an }}San , which can be naturally polarized. One easily checks that the Mumford-Tate group G s := M T X s , s S an G s := M T X s , s ∈ S an  G_(s):=MT_(X_(s)),s inS^("an ")\mathbf{G}_{s}:=\mathbf{M T}_{X_{s}}, s \in S^{\text {an }}Gs:=MTXs,s∈San , is locally constant equal to the so-called generic Mumford-Tate group G G G\mathbf{G}G, outside of a meagre set HL ( S , f ) S an HL ⁡ ( S , f ) ⊂ S an  HL(S,f)subS^("an ")\operatorname{HL}(S, f) \subset S^{\text {an }}HL⁡(S,f)⊂San , the Hodge locus of the morphism f f fff, where it shrinks as exceptional Hodge tensors appear in H B ( X s a n , Z ) H B ∙ X s a n , Z H_(B)^(∙)(X_(s)^(an),Z)H_{\mathrm{B}}^{\bullet}\left(X_{s}^{\mathrm{an}}, \mathbb{Z}\right)HB∙(Xsan,Z). The variation V V V\mathbb{V}V is completely described by its period map
Φ : S an Γ D Φ : S an  → Γ ∖ D Phi:S^("an ")rarr Gamma\\D\Phi: S^{\text {an }} \rightarrow \Gamma \backslash DΦ:San →Γ∖D
Here the period domain D D DDD classifies all possible Z Z Z\mathbb{Z}Z-Hodge structure on the abelian group V Z V Z V_(Z)V_{\mathbb{Z}}VZ, with a fixed polarization and Mumford-Tate group contained in G G G\mathbf{G}G; and Φ Î¦ Phi\PhiΦ maps a point s S an s ∈ S an  s inS^("an ")s \in S^{\text {an }}s∈San  to the point of D D DDD parameterizing the polarized Z Z Z\mathbb{Z}Z-Hodge structure on V Z V Z V_(Z)V_{\mathbb{Z}}VZ defined by V Z , s V Z , s V_(Z,s)\mathbb{V}_{\mathbb{Z}, s}VZ,s (well defined up to the action of the arithmetic group Γ := G G L ( V Z ) Γ := G ∩ G L V Z Gamma:=G nnGL(V_(Z))\Gamma:=G \cap \mathbf{G L}\left(V_{\mathbb{Z}}\right)Γ:=G∩GL(VZ) ).
The transcendence of the comparison isomorphism (1.1) for each fiber X s X s X_(s)X_{s}Xs is embodied in the fact that the Hodge variety Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D is, in general, a mere complex analytic variety
not admitting any algebraic structure; and that the period map Φ Î¦ Phi\PhiΦ is a mere complex analytic map. On the other hand this transcendence is sufficiently constrained so that the following corollary of the Hodge conjecture [96] holds true, as proven by Cattani-Deligne-Kaplan [22]: the Hodge locus H L ( S , f ) H L ( S , f ) HL(S,f)\mathrm{HL}(S, f)HL(S,f) is a countable union of algebraic subvarieties of S S SSS. Remarkably, their result is in fact valid for any polarized Z V H S V Z V H S V ZVHSV\mathbb{Z} \mathrm{VHS} \mathbb{V}ZVHSV on S an S an  S^("an ")S^{\text {an }}San , not necessarily coming from geometry: the Hodge locus HL ( S , V ) HL ⁡ S , V ⊗ HL(S,V^(ox))\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right)HL⁡(S,V⊗) is a countable union of algebraic subvarieties of S S SSS.
In this paper we report on recent advances in the understanding of this interplay between algebraicity and transcendence in Hodge theory, our main object of interest being period maps Φ : S an Γ D Φ : S an  → Γ ∖ D Phi:S^("an ")rarr Gamma\\D\Phi: S^{\text {an }} \rightarrow \Gamma \backslash DΦ:San →Γ∖D. The paper is written for nonexperts: we present the mathematical objects involved, the questions, and the results but give only vague ideas of proofs, if any. It is organized as follows. After Section 2 presenting the objects of Hodge theory (which the advanced reader will skip to refer to on occasion), we present in Section 3 the main driving force behind the recent advances: although period maps are very rarely complex algebraic, their geometry is tame and does not suffer from any of the many possible pathologies of a general holomorphic map. In model-theoretic terms, period maps are definable in the o-minimal structure R a n , e x p R a n , e x p R_(an,exp)\mathbb{R}_{\mathrm{an}, \mathrm{exp}}Ran,exp. In Section 4 , we introduce the general format of bialgebraic structures for comparing the algebraic structure on S S SSS and that on (the compact dual D ˇ D ˇ D^(ˇ)\check{D}Dˇ of) the period domain D D DDD. The heuristic provided by this format, combined with o-minimal geometry, leads to a powerful functional transcendence result: the Ax-Schanuel theorem for polarized Z V H S Z V H S ZVHS\mathbb{Z V H S}ZVHS. It also suggests to interpret variational Hodge theory as a special case of an atypical intersection problem. In Section 5 we describe how this viewpoint leads to a stunning improvement of the result of Cattani, Deligne, and Kaplan: in most cases HL ( S , V ) HL ⁡ S , V ⊗ HL(S,V^(ox))\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right)HL⁡(S,V⊗) is not only a countable union of algebraic varieties, but is actually algebraic on the nose (at least if we restrict to its components of positive period dimension). Finally, in Section 6 we turn briefly to some arithmetic aspects of the theory.
For the sake of simplicity, we focus on the case of pure Hodge structures, only mentioning the references dealing with the mixed case.

2. VARIATIONS OF HODGE STRUCTURES AND PERIOD MAPS

2.1. Polarizable Hodge structures

Let n Z n ∈ Z n inZn \in \mathbb{Z}n∈Z. Let R = Z , Q R = Z , Q R=Z,QR=\mathbb{Z}, \mathbb{Q}R=Z,Q, or R R R\mathbb{R}R. An R R RRR-Hodge structure V V VVV of weight n n nnn is a finitely generated R R RRR-module V R V R V_(R)V_{R}VR together with one of the following equivalent data: a bigrading V C ( := V C ( := V_(C)(:=V_{\mathbb{C}}(:=VC(:= V R R C ) = p + q = n V p , q V R ⊗ R C = ⨁ p + q = n   V p , q {:V_(R)ox_(R)C)=bigoplus_(p+q=n)V^(p,q)\left.V_{R} \otimes_{R} \mathbb{C}\right)=\bigoplus_{p+q=n} V^{p, q}VR⊗RC)=⨁p+q=nVp,q, called the Hodge decomposition, such that V p , q ¯ = V q , p V p , q ¯ = V q , p bar(V^(p,q))=V^(q,p)\overline{V^{p, q}}=V^{q, p}Vp,q¯=Vq,p (the numbers ( dim V p , q ) p + q = n dim ⁡ V p , q p + q = n (dim V^(p,q))_(p+q=n)\left(\operatorname{dim} V^{p, q}\right)_{p+q=n}(dim⁡Vp,q)p+q=n are called the Hodge numbers of V ) V {:V)\left.V\right)V); or a decreasing filtration F F ∙ F^(∙)F^{\bullet}F∙ of V C V C V_(C)V_{\mathbb{C}}VC, called the Hodge filtration, satisfying F p F n + 1 p ¯ = V C F p ⊕ F n + 1 − p ¯ = V C F^(p)o+ bar(F^(n+1-p))=V_(C)F^{p} \oplus \overline{F^{n+1-p}}=V_{\mathbb{C}}Fp⊕Fn+1−p¯=VC. One goes from one to the other through F p = r p V r , n r F p = ⨁ r ≥ p   V r , n − r F^(p)=bigoplus_(r >= p)V^(r,n-r)F^{p}=\bigoplus_{r \geq p} V^{r, n-r}Fp=⨁r≥pVr,n−r and V p , q = F p F q ¯ V p , q = F p ∩ F q ¯ V^(p,q)=F^(p)nn bar(F^(q))V^{p, q}=F^{p} \cap \overline{F^{q}}Vp,q=Fp∩Fq¯. The following group-theoretic description will be most useful to us: a Hodge structure is an R R RRR-module V R V R V_(R)V_{R}VR and a real algebraic representation φ : S G L ( V R ) φ : S → G L V R varphi:SrarrGL(V_(R))\varphi: \mathbf{S} \rightarrow \mathbf{G L}\left(V_{\mathbb{R}}\right)φ:S→GL(VR) whose restriction to G m , R G m , R G_(m,R)\mathbf{G}_{m, \mathbb{R}}Gm,R is defined over Q Q Q\mathbb{Q}Q. Here the Deligne torus S S S\mathbf{S}S denotes the real algebraic group C C ∗ C^(**)\mathbb{C}^{*}C∗ of invertible matrices of the forms ( a b b a ) a − b b a ([a,-b],[b,a])\left(\begin{array}{cc}a & -b \\ b & a\end{array}\right)(a−bba), which contains the diagonal subgroup G m , R G m , R G_(m,R)\mathbf{G}_{m, \mathbb{R}}Gm,R. Being of weight n n nnn is the requirement
that φ G m , R φ ∣ G m , R varphi_(∣G_(m,R))\varphi_{\mid \mathbf{G}_{m, \mathbb{R}}}φ∣Gm,R acts via the character z z n z ↦ z − n z|->z^(-n)z \mapsto z^{-n}z↦z−n. The space V p , q V p , q V^(p,q)V^{p, q}Vp,q is recovered as the eigenspace for the character z z p z z ¯ q z ↦ z − p z z ¯ − q z|->z^(-p_(z)) bar(z)^(-q)z \mapsto z^{-p_{z}} \bar{z}^{-q}z↦z−pzz¯−q of S ( R ) C S ( R ) ≃ C ∗ S(R)≃C^(**)\mathbf{S}(\mathbb{R}) \simeq \mathbb{C}^{*}S(R)≃C∗. A morphism of Hodge structures is a morphism of R R RRR-modules compatible with the bigrading (equivalently, with the Hodge filtration or the S-action).
Example 2.1. We write R ( n ) R ( n ) R(n)R(n)R(n) for the unique R R RRR-Hodge structure of weight 2 n − 2 n -2n-2 n−2n, called the Tate-Hodge structure of weight 2 n − 2 n -2n-2 n−2n, on the rank-one free R R RRR-module ( 2 π i ) n R C ( 2 Ï€ i ) n R ⊂ C (2pii)^(n)R subC(2 \pi \mathrm{i})^{n} R \subset \mathbb{C}(2Ï€i)nR⊂C.
One easily checks that the category of R R RRR-Hodge structures is an abelian category (where the kernels and cokernels coincide with the usual kernels and cokernels in the category of R R RRR-modules, with the induced Hodge filtrations on their complexifications), with natural tensor products V W V ⊗ W V ox WV \otimes WV⊗W and internal homs hom ( V , W ) ( V , W ) (V,W)(V, W)(V,W) (in particular, duals V := V ∨ := V^(vv):=V^{\vee}:=V∨:= hom ( V , R ( 0 ) ) ) ( V , R ( 0 ) ) ) (V,R(0)))(V, R(0)))(V,R(0))). For R = Q R = Q R=QR=\mathbb{Q}R=Q, or R R R\mathbb{R}R, we obtain a Tannakian category, with an obvious exact faithful R R RRR-linear tensor functor ω : ( V R , φ ) V R ω : V R , φ ↦ V R omega:(V_(R),varphi)|->V_(R)\omega:\left(V_{R}, \varphi\right) \mapsto V_{R}ω:(VR,φ)↦VR. In particular, R ( n ) = R ( 1 ) n R ( n ) = R ( 1 ) ⊗ n R(n)=R(1)^(ox n)R(n)=R(1)^{\otimes n}R(n)=R(1)⊗n. If V V VVV is an R R RRR-Hodge structure, we write V ( n ) := V R ( n ) V ( n ) := V ⊗ R ( n ) V(n):=V ox R(n)V(n):=V \otimes R(n)V(n):=V⊗R(n) its n n nnnth Tate twist.
If V = ( V R , φ ) V = V R , φ V=(V_(R),varphi)V=\left(V_{R}, \varphi\right)V=(VR,φ) is an R R RRR-Hodge structure of weight n n nnn, a polarization for V V VVV is a morphism of R R RRR-Hodge structures q : V 2 R ( n ) q : V ⊗ 2 → R ( − n ) q:V^(ox2)rarr R(-n)q: V^{\otimes 2} \rightarrow R(-n)q:V⊗2→R(−n) such that ( 2 π i ) n q ( x , φ ( i ) y ) ( 2 Ï€ i ) n q ( x , φ ( i ) y ) (2pii)^(n)q(x,varphi(i)y)(2 \pi \mathrm{i})^{n} q(x, \varphi(\mathrm{i}) y)(2Ï€i)nq(x,φ(i)y) is a positivedefinite bilinear form on V R V R V_(R)V_{\mathbb{R}}VR, called the Hodge form associated with the polarization. If there exists a polarization for V V VVV then V V VVV is said polarizable. One easily checks that the category of polarizable Q Q Q\mathbb{Q}Q-Hodge structures is semisimple.
Example 2.2. Let M M MMM be a compact complex manifold. If M M MMM admits a Kähler metric, the singular cohomology H B n ( M , Z ) H B n ( M , Z ) H_(B)^(n)(M,Z)H_{\mathrm{B}}^{n}(M, \mathbb{Z})HBn(M,Z) is naturally a Z Z Z\mathbb{Z}Z-Hodge structure of weight n n nnn, see [52], [94, CHAP. 6]:
H B n ( M , Z ) Z C = H d R n ( M , C ) = p + q = n H p , q ( M ) H B n ( M , Z ) ⊗ Z C = H d R n ( M , C ) = ⨁ p + q = n   H p , q ( M ) H_(B)^(n)(M,Z)oxZC=H_(dR)^(n)(M,C)=bigoplus_(p+q=n)H^(p,q)(M)H_{\mathrm{B}}^{n}(M, \mathbb{Z}) \otimes \mathbb{Z} \mathbb{C}=H_{d R}^{n}(M, \mathbb{C})=\bigoplus_{p+q=n} H^{p, q}(M)HBn(M,Z)⊗ZC=HdRn(M,C)=⨁p+q=nHp,q(M)
where H d R ( M , C ) H d R ∙ ( M , C ) H_(dR)^(∙)(M,C)H_{d R}^{\bullet}(M, \mathbb{C})HdR∙(M,C) denotes the de Rham cohomology of the complex ( A ( M , C ) , d ) A ∙ ( M , C ) , d (A^(∙)(M,C),d)\left(A^{\bullet}(M, \mathbb{C}), d\right)(A∙(M,C),d) of C C ∞ C^(oo)\mathrm{C}^{\infty}C∞ differential forms on M M MMM, the first equality is the canonical isomorphism obtained by integrating forms on cycles (de Rham theorem), and the complex vector subspace H p , q ( M ) H p , q ( M ) H^(p,q)(M)H^{p, q}(M)Hp,q(M) of H d R n ( M , C ) H d R n ( M , C ) H_(dR)^(n)(M,C)H_{\mathrm{dR}}^{n}(M, \mathbb{C})HdRn(M,C) is generated by the d d ddd-closed forms of type ( p , q ) ( p , q ) (p,q)(p, q)(p,q), and thus satisfies automatically H p , q ( M ) ¯ = H q , p ( M ) H p , q ( M ) ¯ = H q , p ( M ) bar(H^(p,q)(M))=H^(q,p)(M)\overline{H^{p, q}(M)}=H^{q, p}(M)Hp,q(M)¯=Hq,p(M). Although the second equality depends only on the complex structure on M M MMM, its proof relies on the choice of a Kähler form ω ω omega\omegaω on M M MMM through the following sequence of isomorphisms:
H d R n ( M , C ) H Δ ω n ( M ) = p + q = n H Δ ω p , q ( M ) p + q = n H p , q ( M ) H d R n ( M , C ) → ∼ H Δ ω n ( M ) = ⨁ p + q = n   H Δ ω p , q ( M ) → ∼ ⨁ p + q = n   H p , q ( M ) H_(dR)^(n)(M,C)rarr"∼"H_(Delta_(omega))^(n)(M)=bigoplus_(p+q=n)H_(Delta_(omega))^(p,q)(M)rarr"∼"bigoplus_(p+q=n)H^(p,q)(M)H_{d R}^{n}(M, \mathbb{C}) \xrightarrow{\sim} \mathscr{H}_{\Delta_{\omega}}^{n}(M)=\bigoplus_{p+q=n} \mathscr{H}_{\Delta_{\omega}}^{p, q}(M) \xrightarrow{\sim} \bigoplus_{p+q=n} H^{p, q}(M)HdRn(M,C)→∼HΔωn(M)=⨁p+q=nHΔωp,q(M)→∼⨁p+q=nHp,q(M)
where H Δ ω n ( M ) H Δ ω n ( M ) H_(Delta_(omega))^(n)(M)\mathscr{H}_{\Delta_{\omega}}^{n}(M)HΔωn(M) denotes the vector space of Δ ω Δ ω Delta_(omega)\Delta_{\omega}Δω-harmonic differential forms on M M MMM and H Δ ω p , q ( M ) H Δ ω p , q ( M ) H_(Delta_(omega))^(p,q)(M)\mathscr{H}_{\Delta_{\omega}}^{p, q}(M)HΔωp,q(M) its subspace of Δ ω Δ ω Delta_(omega)\Delta_{\omega}Δω-harmonic ( p , q ) ( p , q ) (p,q)(p, q)(p,q)-forms. The heart of Hodge theory is thus reduced to the statement that the Laplacian Δ ω Δ ω Delta_(omega)\Delta_{\omega}Δω of a Kähler metric preserves the type of forms. The choice of a Kähler form ω ω omega\omegaω on M M MMM also defines, through the hard Lefschetz theorem [94, THEOREM 6.25], a polarization of the R R R\mathbb{R}R-Hodge structure H n ( M , R ) H n ( M , R ) H^(n)(M,R)H^{n}(M, \mathbb{R})Hn(M,R), see [94, THEOREM 6.32]. If f : M N f : M → N f:M rarr Nf: M \rightarrow Nf:M→N is any holomorphic map between compact complex manifolds admitting Kähler metrics then both f : H B n ( N , Z ) H B n ( M , Z ) f ∗ : H B n ( N , Z ) → H B n ( M , Z ) f^(**):H_(B)^(n)(N,Z)rarrH_(B)^(n)(M,Z)f^{*}: H_{\mathrm{B}}^{n}(N, \mathbb{Z}) \rightarrow H_{\mathrm{B}}^{n}(M, \mathbb{Z})f∗:HBn(N,Z)→HBn(M,Z) and the Gysin morphism f : H B n ( M , Z ) f ∗ : H B n ( M , Z ) → f_(**):H_(B)^(n)(M,Z)rarrf_{*}: H_{\mathrm{B}}^{n}(M, \mathbb{Z}) \rightarrowf∗:HBn(M,Z)→ H B n 2 r ( N , Z ) ( r ) H B n − 2 r ( N , Z ) ( − r ) H_(B)^(n-2r)(N,Z)(-r)H_{\mathrm{B}}^{n-2 r}(N, \mathbb{Z})(-r)HBn−2r(N,Z)(−r) are morphism of Z Z Z\mathbb{Z}Z-Hodge structures, where r = dim M dim N r = dim ⁡ M − dim ⁡ N r=dim M-dim Nr=\operatorname{dim} M-\operatorname{dim} Nr=dim⁡M−dim⁡N.
Example 2.3. Suppose moreover that M = X an M = X an  M=X^("an ")M=X^{\text {an }}M=Xan  is the compact complex manifold analytification of a smooth projective variety X X XXX over C C C\mathbb{C}C. In that case, H B n ( X , Z ) H B n ( X , Z ) H_(B)^(n)(X,Z)H_{\mathrm{B}}^{n}(X, \mathbb{Z})HBn(X,Z) is a polarizable Z Z Z\mathbb{Z}Z-Hodge structure. Indeed, the Kähler class [ ω ] [ ω ] [omega][\omega][ω] can be chosen as the first Chern class of an ample line bundle on X X XXX, giving rise to a rational Lefschetz decomposition and (after clearing denominators by multiplying by a sufficiently large integer) to an integral polarization. Moreover, the Hodge filtration F F ∙ F^(∙)F^{\bullet}F∙ on H B n ( X an , C ) H B n X an  , C H_(B)^(n)(X^("an "),C)H_{\mathrm{B}}^{n}\left(X^{\text {an }}, \mathbb{C}\right)HBn(Xan ,C) can be defined algebraically: upon identifying H B n ( X an , C ) H B n X an  , C H_(B)^(n)(X^("an "),C)H_{\mathrm{B}}^{n}\left(X^{\text {an }}, \mathbb{C}\right)HBn(Xan ,C) with the algebraic de Rham cohomology H d R n ( X / C ) := H n ( X , Ω X / C ) H d R n ( X / C ) := H n X , Ω X / C ∙ H_(dR)^(n)(X//C):=H^(n)(X,Omega_(X//C)^(∙))H_{\mathrm{dR}}^{n}(X / \mathbb{C}):=H^{n}\left(X, \Omega_{X / \mathbb{C}}^{\bullet}\right)HdRn(X/C):=Hn(X,ΩX/C∙), the Hodge filtration is given by F p = Im ( H n ( X , Ω X / C p ) H B n ( X an , C ) ) F p = Im ⁡ H n X , Ω X / C ∙ ≥ p → H B n X an  , C F^(p)=Im(H^(n)(X,Omega_(X//C)^(∙ >= p))rarrH_(B)^(n)(X^("an "),C))F^{p}=\operatorname{Im}\left(H^{n}\left(X, \Omega_{X / \mathbb{C}}^{\bullet \geq p}\right) \rightarrow H_{\mathrm{B}}^{n}\left(X^{\text {an }}, \mathbb{C}\right)\right)Fp=Im⁡(Hn(X,ΩX/C∙≥p)→HBn(Xan ,C)). It follows that if X X XXX is defined over a subfield K K KKK of C C C\mathbb{C}C, then the Hodge filtration F F ∙ F^(∙)F^{\bullet}F∙ on H B n ( X an , C ) = H B n X an  , C = H_(B)^(n)(X^("an "),C)=H_{\mathrm{B}}^{n}\left(X^{\text {an }}, \mathbb{C}\right)=HBn(Xan ,C)= H d R n ( X / K ) K C H d R n ( X / K ) ⊗ K C H_(dR)^(n)(X//K)ox_(K)CH_{\mathrm{dR}}^{n}(X / K) \otimes_{K} \mathbb{C}HdRn(X/K)⊗KC is defined over K K KKK.
Example 2.4. The functor which assigns to a complex abelian variety A A AAA its H B 1 ( A an , Z ) H B 1 A an  , Z H_(B)^(1)(A^("an "),Z)H_{\mathrm{B}}^{1}\left(A^{\text {an }}, \mathbb{Z}\right)HB1(Aan ,Z) defines an equivalence of categories between abelian varieties and polarizable Z Z Z\mathbb{Z}Z-Hodge structures of weight 1 and type ( 1 , 0 ) ( 1 , 0 ) (1,0)(1,0)(1,0) and ( 0 , 1 ) ( 0 , 1 ) (0,1)(0,1)(0,1).

2.2. Hodge classes and Mumford-Tate group

Let R = Z R = Z R=ZR=\mathbb{Z}R=Z or Q Q Q\mathbb{Q}Q and let V V VVV be an R R RRR-Hodge structure. A Hodge class for V V VVV is a vector in V 0 , 0 V Q = F 0 V C V Q V 0 , 0 ∩ V Q = F 0 V C ∩ V Q V^(0,0)nnV_(Q)=F^(0)V_(C)nnV_(Q)V^{0,0} \cap V_{\mathbb{Q}}=F^{0} V_{\mathbb{C}} \cap V_{\mathbb{Q}}V0,0∩VQ=F0VC∩VQ. For instance, any morphism of R R RRR-Hodge structures f : V W f : V → W f:V rarr Wf: V \rightarrow Wf:V→W defines a Hodge class in the internal hom ( V , W ) hom ⁡ ( V , W ) hom(V,W)\operatorname{hom}(V, W)hom⁡(V,W). Let T m , n V Q T m , n V Q T^(m,n)V_(Q)T^{m, n} V_{\mathbb{Q}}Tm,nVQ denote the Q Q Q\mathbb{Q}Q Hodge structure V Q m hom ( V , R ( 0 ) ) Q n V Q ⊗ m ⊗ hom ⁡ ( V , R ( 0 ) ) Q ⊗ n V_(Q)^(ox m)ox hom(V,R(0))_(Q)^(ox n)V_{\mathbb{Q}}^{\otimes m} \otimes \operatorname{hom}(V, R(0))_{\mathbb{Q}}^{\otimes n}VQ⊗m⊗hom⁡(V,R(0))Q⊗n. A Hodge tensor for V V VVV is a Hodge class in some T m , n V Q T m , n V Q T^(m,n)V_(Q)T^{m, n} V_{\mathbb{Q}}Tm,nVQ.
The main invariant of an R R RRR-Hodge structure is its Mumford-Tate group. For any R R RRR Hodge structure V V VVV we denote by V ⟨ V ⟩ (:V:)\langle V\rangle⟨V⟩ the Tannakian subcategory of the category of Q Q Q\mathbb{Q}Q-Hodge structures generated by V Q V Q V_(Q)V_{\mathbb{Q}}VQ; in other words, V ⟨ V ⟩ (:V:)\langle V\rangle⟨V⟩ is the smallest full subcategory containing V , Q ( 0 ) V , Q ( 0 ) V,Q(0)V, \mathbb{Q}(0)V,Q(0) and stable under , ⊕ , ⊗ o+,ox\oplus, \otimes⊕,⊗, and taking subquotients. If ω V ω V omega_(V)\omega_{V}ωV denotes the restriction of the tensor functor ω ω omega\omegaω to V ⟨ V ⟩ (:V:)\langle V\rangle⟨V⟩, the functor Aut ( ω V ) Aut ⊗ ⁡ ω V Aut^(ox)(omega_(V))\operatorname{Aut}^{\otimes}\left(\omega_{V}\right)Aut⊗⁡(ωV) is representable by some closed Q Q Q\mathbb{Q}Q-algebraic subgroup G V G L ( V Q ) G V ⊂ G L V Q G_(V)subGL(V_(Q))\mathbf{G}_{V} \subset \mathbf{G L}\left(V_{\mathbb{Q}}\right)GV⊂GL(VQ), called the Mumford-Tate group of V V VVV, and ω V ω V omega_(V)\omega_{V}ωV defines an equivalence of categories V Rep Q G V ⟨ V ⟩ ≃ Rep Q ⁡ G V (:V:)≃Rep_(Q)G_(V)\langle V\rangle \simeq \operatorname{Rep}_{\mathbb{Q}} \mathbf{G}_{V}⟨V⟩≃RepQ⁡GV. See [33, II, 2.11].
The Mumford-Tate group G V G V G_(V)\mathbf{G}_{V}GV can also be characterized as the fixator in G L ( V Q ) G L V Q GL(V_(Q))\mathbf{G L}\left(V_{\mathbb{Q}}\right)GL(VQ) of the Hodge tensors for V V VVV, or equivalently, writing V = ( V R , φ ) V = V R , φ V=(V_(R),varphi)V=\left(V_{R}, \varphi\right)V=(VR,φ), as the smallest Q Q Q\mathbb{Q}Q-algebraic subgroup of G L ( V Q ) G L V Q GL(V_(Q))\mathbf{G L}\left(V_{\mathbb{Q}}\right)GL(VQ) whose base change to R R R\mathbb{R}R contains the image Im φ Im ⁡ φ Im varphi\operatorname{Im} \varphiIm⁡φ. In particular φ φ varphi\varphiφ factorizes as φ : S G V , R φ : S → G V , R varphi:SrarrG_(V,R)\varphi: \mathbf{S} \rightarrow \mathbf{G}_{V, \mathbb{R}}φ:S→GV,R. The group G V G V G_(V)\mathbf{G}_{V}GV is thus connected, and reductive if V V VVV is polarizable. See [2, LEMMA 2].
Example 2.5. G Z ( n ) = G m G Z ( n ) = G m G_(Z(n))=G_(m)\mathbf{G}_{\mathbb{Z}(n)}=\mathbf{G}_{m}GZ(n)=Gm if n 0 n ≠ 0 n!=0n \neq 0n≠0 and G Z ( 0 ) = { 1 } G Z ( 0 ) = { 1 } G_(Z(0))={1}\mathbf{G}_{\mathbb{Z}(0)}=\{1\}GZ(0)={1}.
Example 2.6. Let A A AAA be a complex abelian variety and let V := H B 1 ( A an , Z ) V := H B 1 A an  , Z V:=H_(B)^(1)(A^("an "),Z)V:=H_{\mathrm{B}}^{1}\left(A^{\text {an }}, \mathbb{Z}\right)V:=HB1(Aan ,Z) be the associated Z Z Z\mathbb{Z}Z-Hodge structure of weight 1 . We write G A := G V G A := G V G_(A):=G_(V)\mathbf{G}_{A}:=\mathbf{G}_{V}GA:=GV. The choice of an ample line bundle on A A AAA defines a polarization q q qqq on V V VVV. On the one hand, the endomorphism algebra D := D := D:=D:=D:= End 0 ( A ) ( := End ( A ) Z Q ) End 0 ⁡ ( A ) := End ⁡ ( A ) ⊗ Z Q End^(0)(A)(:=End(A)ox_(Z)Q)\operatorname{End}^{0}(A)\left(:=\operatorname{End}(A) \otimes_{\mathbb{Z}} \mathbb{Q}\right)End0⁡(A)(:=End⁡(A)⊗ZQ) is a finite-dimensional semisimple Q Q Q\mathbb{Q}Q-algebra which, in view of Example 2.4, identifies with End ( V Q ) G A End ⁡ V Q G A End (V_(Q))^(G_(A))\operatorname{End}\left(V_{\mathbb{Q}}\right)^{\mathbf{G}_{A}}End⁡(VQ)GA. Thus G A G L D ( V Q ) G A ⊂ G L D V Q G_(A)subGL_(D)(V_(Q))\mathbf{G}_{A} \subset \mathbf{G L}_{D}\left(V_{\mathbb{Q}}\right)GA⊂GLD(VQ). On the other hand, the polarization q q qqq defines a Hodge class in hom ( V Q 2 , Q ( 1 ) ) hom ⁡ V Q ⊗ 2 , Q ( − 1 ) hom(V_(Q)^(ox2),Q(-1))\operatorname{hom}\left(V_{\mathbb{Q}}^{\otimes 2}, \mathbb{Q}(-1)\right)hom⁡(VQ⊗2,Q(−1)) thus G A G A G_(A)\mathbf{G}_{A}GA has to be contained in
the group G S p ( V Q , q ) G S p V Q , q GSp(V_(Q),q)\mathbf{G S p}\left(V_{\mathbb{Q}}, q\right)GSp(VQ,q) of symplectic similitudes of V Q V Q V_(Q)V_{\mathbb{Q}}VQ with respect to the symplectic form q q qqq. Finally, G A G L D ( V Q ) G S p ( V Q , q ) G A ⊂ G L D V Q ∩ G S p V Q , q G_(A)subGL_(D)(V_(Q))nnGSp(V_(Q),q)\mathbf{G}_{A} \subset \mathbf{G L}_{D}\left(V_{\mathbb{Q}}\right) \cap \mathbf{G S p}\left(V_{\mathbb{Q}}, q\right)GA⊂GLD(VQ)∩GSp(VQ,q).
If A = E A = E A=EA=EA=E is an elliptic curve, it follows readily that either D = Q D = Q D=QD=\mathbb{Q}D=Q and G E = G L 2 G E = G L 2 G_(E)=GL_(2)\mathbf{G}_{E}=\mathbf{G L}_{2}GE=GL2, or D D DDD is an imaginary quadratic field ( E E EEE has complex multiplication) and G E = T D G E = T D G_(E)=T_(D)\mathbf{G}_{E}=\mathbf{T}_{D}GE=TD, the Q Q Q\mathbb{Q}Q-torus defined by T D ( S ) = ( D Q S ) T D ( S ) = ( D ⊗ Q S ) ∗ T_(D)(S)=(D oxQS)^(**)\mathbf{T}_{D}(S)=(D \otimes \mathbb{Q} S)^{*}TD(S)=(D⊗QS)∗ for any Q Q Q\mathbb{Q}Q-algebra S S SSS.

2.3. Period domains and Hodge data

Let V Z V Z V_(Z)V_{\mathbb{Z}}VZ be a finitely generated abelian group V Z V Z V_(Z)V_{\mathbb{Z}}VZ of rank r r rrr. Fix a positive integer n n nnn, a ( 1 ) n ( − 1 ) n (-1)^(n)(-1)^{n}(−1)n-symmetric bilinear form q Z q Z q_(Z)q_{\mathbf{Z}}qZ on V Z V Z V_(Z)V_{\mathbb{Z}}VZ and a collection of nonnegative integers ( h p , q ) ( p , q 0 , p + q = n ) h p , q ( p , q ≥ 0 , p + q = n ) (h^(p,q))(p,q >= 0,p+q=n)\left(h^{p, q}\right)(p, q \geq 0, p+q=n)(hp,q)(p,q≥0,p+q=n) such that h p , q = h q , p h p , q = h q , p h^(p,q)=h^(q,p)h^{p, q}=h^{q, p}hp,q=hq,p and h p , q = r ∑ h p , q = r sumh^(p,q)=r\sum h^{p, q}=r∑hp,q=r. Associated with ( n , q Z , ( h p , q ) n , q Z , h p , q (n,q_(Z),(h^(p,q)):}\left(n, q_{\mathbb{Z}},\left(h^{p, q}\right)\right.(n,qZ,(hp,q) ) we want to define a period domain D D DDD classifying Z Z Z\mathbb{Z}Z-Hodge structures of weight n n nnn on V Z V Z V_(Z)V_{\mathbb{Z}}VZ, polarized by q Z q Z q_(Z)q_{\mathbb{Z}}qZ, and with Hodge numbers h p , q h p , q h^(p,q)h^{p, q}hp,q. Setting f p = r p h r , n r f p = ∑ r ≥ p   h r , n − r f^(p)=sum_(r >= p)h^(r,n-r)f^{p}=\sum_{r \geq p} h^{r, n-r}fp=∑r≥phr,n−r, we first define the compact dual D ˇ D ˇ D^(ˇ)\check{D}Dˇ parametrizing the finite decreasing filtrations F F ∙ F^(∙)F^{\bullet}F∙ on V C V C V_(C)V_{\mathbb{C}}VC satisfying ( F p ) q Z = F n + 1 p F p ⊥ q Z = F n + 1 − p (F^(p))^(_|__(q_(Z)))=F^(n+1-p)\left(F^{p}\right)^{\perp_{q_{\mathbb{Z}}}}=F^{n+1-p}(Fp)⊥qZ=Fn+1−p and dim F p = f p dim ⁡ F p = f p dim F^(p)=f^(p)\operatorname{dim} F^{p}=f^{p}dim⁡Fp=fp. This is a closed algebraic subvariety of the product of Grassmannians p Gr ( f p , V C ) ∏ p   Gr ⁡ f p , V C prod_(p)Gr(f^(p),V_(C))\prod_{p} \operatorname{Gr}\left(f^{p}, V_{\mathbb{C}}\right)∏pGr⁡(fp,VC). The period domain D D ˇ an D ⊂ D ˇ an  D subD^(ˇ)^("an ")D \subset \check{D}^{\text {an }}D⊂Dˇan  is the open subset where the Hodge form is positive definite. If G := GAut ( V Q , q Q ) G := GAut ⁡ V Q , q Q G:=GAut(V_(Q),q_(Q))\mathbf{G}:=\operatorname{GAut}\left(V_{\mathbb{Q}}, q_{\mathbb{Q}}\right)G:=GAut⁡(VQ,qQ) denotes the group of similitudes of q Q q Q q_(Q)q_{\mathbb{Q}}qQ, one easily checks that G ( C ) G ( C ) G(C)\mathbf{G}(\mathbb{C})G(C) acts transitively on D ˇ an D ˇ an  D^(ˇ)^("an ")\check{D}^{\text {an }}Dˇan , which is thus a flag variety for G C G C G_(C)\mathbf{G}_{\mathbb{C}}GC; and that the connected component G := G der ( R ) + G := G der  ( R ) + G:=G^("der ")(R)^(+)G:=\mathbf{G}^{\text {der }}(\mathbb{R})^{+}G:=Gder (R)+of the identity in the derived group G der ( R ) G der  ( R ) G^("der ")(R)\mathbf{G}^{\text {der }}(\mathbb{R})Gder (R) acts transitively on D D DDD, which identifies with an open G G GGG-orbit in D ˇ D ˇ D^(ˇ)\check{D}Dˇ. If we fix a base point o D o ∈ D o in Do \in Do∈D and denote by P P PPP and M M MMM its stabilizer in G ( C ) G ( C ) G(C)\mathbf{G}(\mathbb{C})G(C) and G G GGG, respectively, the period domain D D DDD is thus the homogeneous space
D = G / M D ˇ an = G ( C ) / P . D = G / M ↪ D ˇ an  = G ( C ) / P .  D=G//M↪D^(ˇ)^("an ")=G(C)//P". "D=G / M \hookrightarrow \check{D}^{\text {an }}=\mathbf{G}(\mathbb{C}) / P \text {. }D=G/M↪Dˇan =G(C)/P. 
The group P P PPP is a parabolic subgroup of G ( C ) G ( C ) G(C)\mathbf{G}(\mathbb{C})G(C). Its subgroup M = P G M = P ∩ G M=P nn GM=P \cap GM=P∩G, consisting of real elements, not only fixes the filtration F o F o ∙ F_(o)^(∙)F_{o}^{\bullet}Fo∙ but also the Hodge decomposition, hence the Hodge form, at o o ooo. It is thus a compact subgroup of G G GGG and D D DDD is an open elliptic orbit of G G GGG in D ˇ D ˇ D^(ˇ)\check{D}Dˇ.
Example 2.7. Let n = 1 n = 1 n=1n=1n=1, suppose that the only nonzero Hodge numbers are h 1 , 0 = h 0 , 1 = g h 1 , 0 = h 0 , 1 = g h^(1,0)=h^(0,1)=gh^{1,0}=h^{0,1}=gh1,0=h0,1=g, q Z q Z q_(Z)q_{\mathbb{Z}}qZ is a symplectic form and D D DDD is the subset of Gr ( g , V C ) Gr ⁡ g , V C Gr(g,V_(C))\operatorname{Gr}\left(g, V_{\mathbb{C}}\right)Gr⁡(g,VC) consisting of q C q C q_(C)q_{\mathbb{C}}qC-Lagrangian subspaces F 1 F 1 F^(1)F^{1}F1 on which i q C ( u , u ¯ ) i q C ( u , u ¯ ) iq_(C)(u, bar(u))\mathrm{i} q_{\mathbb{C}}(u, \bar{u})iqC(u,u¯) is positive definite. In this case G = G S p 2 g , G = S p 2 g ( R ) G = G S p 2 g , G = S p 2 g ( R ) G=GSp_(2g),G=Sp_(2g)(R)\mathbf{G}=\mathbf{G S p}_{2 g}, G=\mathbf{S p}_{2 g}(\mathbb{R})G=GSp2g,G=Sp2g(R), M = S O 2 g ( R ) M = S O 2 g ( R ) M=SO_(2g)(R)M=\mathbf{S O}_{2 g}(\mathbb{R})M=SO2g(R) is a maximal compact subgroup of the connected Lie group G G GGG, and D = D = D=D=D= G / M G / M G//MG / MG/M is a bounded symmetric domain naturally biholomorphic to Siegel's upper half-space S g S g S_(g)\mathfrak{S}_{g}Sg of g × g g × g g xx gg \times gg×g-complex symmetric matrices Z = X + i Y Z = X + i Y Z=X+iYZ=X+\mathrm{i} YZ=X+iY with Y Y YYY positive definite. When g = 1 , D g = 1 , D g=1,Dg=1, Dg=1,D is the Poincaré disk, biholomorphic to the Poincaré upper half-space S S S\mathfrak{S}S.
More generally, let G G G\mathbf{G}G be a connected reductive Q Q Q\mathbb{Q}Q-algebraic group and let φ : S φ : S → varphi:Srarr\varphi: \mathbf{S} \rightarrowφ:S→ G R G R G_(R)\mathbf{G}_{\mathbb{R}}GR be a real algebraic morphism such that φ G m , R φ ∣ G m , R varphi_(∣G_(m,R))\varphi_{\mid \mathbf{G}_{m, \mathbb{R}}}φ∣Gm,R is defined over Q Q Q\mathbb{Q}Q. We assume that G G G\mathbf{G}G is the Mumford-Tate group of φ φ varphi\varphiφ. The period domain (or Hodge domain) D D DDD associated with φ : S G R φ : S → G R varphi:SrarrG_(R)\varphi: \mathbf{S} \rightarrow \mathbf{G}_{\mathbb{R}}φ:S→GR is the connected component of the G ( R ) G ( R ) G(R)\mathbf{G}(\mathbb{R})G(R)-conjugacy class of φ : S G R φ : S → G R varphi:SrarrG_(R)\varphi: \mathbf{S} \rightarrow \mathbf{G}_{\mathbb{R}}φ:S→GR in Hom ( S , G R ) Hom ⁡ S , G R Hom(S,G_(R))\operatorname{Hom}\left(\mathbf{S}, \mathbf{G}_{\mathbb{R}}\right)Hom⁡(S,GR). Again, one easily checks that D D DDD is an open elliptic orbit of G := G d e r ( R ) + G := G d e r ( R ) + G:=G^(der)(R)^(+)G:=\mathbf{G}^{\mathrm{der}}(\mathbb{R})^{+}G:=Gder(R)+ in the compact dual flag variety D ˇ an D ˇ an  D^(ˇ)^("an ")\check{D}^{\text {an }}Dˇan , the G ( C ) G ( C ) G(C)\mathbf{G}(\mathbb{C})G(C)-conjugacy class of φ C μ : G m , C G C φ C ∘ μ : G m , C → G C varphi_(C)@mu:G_(m,C)rarrG_(C)\varphi_{\mathbb{C}} \circ \mu: \mathbf{G}_{\mathrm{m}, \mathbb{C}} \rightarrow \mathbf{G}_{\mathbb{C}}φC∘μ:Gm,C→GC, where μ : G m , C S C = G m , C × G m , C μ : G m , C → S C = G m , C × G m , C mu:G_(m,C)rarrS_(C)=G_(m,C)xxG_(m,C)\mu: \mathbf{G}_{\mathbf{m}, \mathbb{C}} \rightarrow \mathbf{S}_{\mathbb{C}}=\mathbf{G}_{\mathbf{m}, \mathbb{C}} \times \mathbf{G}_{\mathbf{m}, \mathbb{C}}μ:Gm,C→SC=Gm,C×Gm,C is the cocharacter z ( z , 1 ) z ↦ ( z , 1 ) z|->(z,1)z \mapsto(z, 1)z↦(z,1). See [41] for details. The pair ( G , D ) ( G , D ) (G,D)(\mathbf{G}, D)(G,D) is called a (connected) Hodge datum. A morphism of Hodge data ( G , D ) ( G , D ) → (G,D)rarr(\mathbf{G}, D) \rightarrow(G,D)→
( G , D ) G ′ , D ′ (G^('),D^('))\left(\mathbf{G}^{\prime}, D^{\prime}\right)(G′,D′) is a morphism ρ : G G ρ : G → G ′ rho:GrarrG^(')\rho: \mathbf{G} \rightarrow \mathbf{G}^{\prime}ρ:G→G′ sending D D DDD to D D ′ D^(')D^{\prime}D′. Any linear representation λ : G λ : G → lambda:Grarr\lambda: \mathbf{G} \rightarrowλ:G→ G L ( V Q ) G L V Q GL(V_(Q))\mathbf{G L}\left(V_{\mathbb{Q}}\right)GL(VQ) defines a G ( Q ) G ( Q ) G(Q)\mathbf{G}(\mathbb{Q})G(Q)-equivariant local system V ˇ λ V ˇ λ V^(ˇ)_(lambda)\check{\mathbb{V}}_{\lambda}Vˇλ on D ˇ an D ˇ an  D^(ˇ)^("an ")\check{D}^{\text {an }}Dˇan . Moreover, each point x x ∈ x inx \inx∈ D D DDD, seen as a morphism φ x : S G R φ x : S → G R varphi_(x):SrarrG_(R)\varphi_{x}: \mathbf{S} \rightarrow \mathbf{G}_{\mathbb{R}}φx:S→GR, defines a Q Q Q\mathbb{Q}Q-Hodge structure V x := ( V Q , λ φ x ) V x := V Q , λ ∘ φ x V_(x):=(V_(Q),lambda@varphi_(x))V_{x}:=\left(V_{\mathbb{Q}}, \lambda \circ \varphi_{x}\right)Vx:=(VQ,λ∘φx). The G ( C ) G ( C ) G(C)\mathbf{G}(\mathbb{C})G(C)-equivariant filtration F V ˇ λ := G ad ( C ) × P , λ F V o , C F ∙ V ˇ λ := G ad  ( C ) × P , λ F ∙ V o , C F^(∙)V^(ˇ)_(lambda):=G^("ad ")(C)xx_(P,lambda)F^(∙)V_(o,C)F^{\bullet} \check{\mathcal{V}}_{\lambda}:=\mathbf{G}^{\text {ad }}(\mathbb{C}) \times_{P, \lambda} F^{\bullet} V_{o, \mathbb{C}}F∙Vˇλ:=Gad (C)×P,λF∙Vo,C of the holomorphic vector bundle V ˇ λ := G ad ( C ) × P , λ V o , C V ˇ λ := G ad  ( C ) × P , λ V o , C V^(ˇ)_(lambda):=G^("ad ")(C)xx_(P,lambda)V_(o,C)\check{\mathcal{V}}_{\lambda}:=\mathbf{G}^{\text {ad }}(\mathbb{C}) \times_{P, \lambda} V_{o, \mathbb{C}}Vˇλ:=Gad (C)×P,λVo,C on D ˇ an D ˇ an  D^(ˇ)^("an ")\check{D}^{\text {an }}Dˇan  induces the Hodge filtration on V x V x V_(x)V_{x}Vx for each x D x ∈ D x in Dx \in Dx∈D. The Mumford-Tate group of V x V x V_(x)V_{x}Vx is G G G\mathbf{G}G precisely when x D τ ( D ) x ∈ D ∖ ⋃ Ï„ D ′ x in D\\uuu tau(D^('))x \in D \backslash \bigcup \tau\left(D^{\prime}\right)x∈D∖⋃τ(D′), where τ Ï„ tau\tauÏ„ ranges through the countable set of morphisms of Hodge data τ : ( G , D ) τ ( G , D ) Ï„ : G ′ , D ′ Ï„ → ( G , D ) tau:(G^('),D^('))^(tau)rarr(G,D)\tau:\left(\mathbf{G}^{\prime}, D^{\prime}\right)^{\tau} \rightarrow(\mathbf{G}, D)Ï„:(G′,D′)τ→(G,D). The complex analytic subvarieties τ ( D ) Ï„ D ′ tau(D^('))\tau\left(D^{\prime}\right)Ï„(D′) of D D DDD are called the special subvarieties of D D DDD.
The following geometric feature of D ˇ D ˇ D^(ˇ)\check{D}Dˇ will be crucial for us. The algebraic tangent bundle T D ˇ T D ˇ TD^(ˇ)T \check{D}TDˇ naturally identifies, as a G C G C G_(C)\mathbf{G}_{\mathbb{C}}GC-equivariant bundle, with the quotient vector bundle V ˇ Ad / F 0 V ˇ Ad V ˇ Ad  / F 0 V ˇ Ad  V^(ˇ)_("Ad ")//F^(0)V^(ˇ)_("Ad ")\check{\mathcal{V}}_{\text {Ad }} / F^{0} \check{\mathcal{V}}_{\text {Ad }}VˇAd /F0VˇAd , where A d : G G L ( g ) A d : G → G L ( g ) Ad:GrarrGL(g)\mathrm{Ad}: \mathbf{G} \rightarrow \mathbf{G L}(\mathrm{g})Ad:G→GL(g) is the adjoint representation on the Lie algebra g g g\mathrm{g}g of G G G\mathbf{G}G In particular, it is naturally filtered by the F i T D ˇ := F i V ˇ A d / F 0 V ˇ A d , i 1 F i T D ˇ := F i V ˇ A d / F 0 V ˇ A d , i ≤ − 1 F^(i)TD^(ˇ):=F^(i)V^(ˇ)_(Ad)//F^(0)V^(ˇ)_(Ad),i <= -1F^{i} T \check{D}:=F^{i} \check{\mathcal{V}}_{\mathrm{Ad}} / F^{0} \check{\mathcal{V}}_{\mathrm{Ad}}, i \leq-1FiTDˇ:=FiVˇAd/F0VˇAd,i≤−1. The subbundle F 1 T D ˇ F − 1 T D ˇ F^(-1)TD^(ˇ)F^{-1} T \check{D}F−1TDˇ is called the horizontal tangent bundle of D ˇ D ˇ D^(ˇ)\check{D}Dˇ.

2.4. Hodge varieties

Let ( G , D ) ( G , D ) (G,D)(\mathbf{G}, D)(G,D) be a Hodge datum as in Section 2.3. A Hodge variety is the quotient Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D of D D DDD by an arithmetic lattice Γ Î“ Gamma\GammaΓ of G ( Q ) + := G ( Q ) G G ( Q ) + := G ( Q ) ∩ G G(Q)^(+):=G(Q)nn G\mathbf{G}(\mathbb{Q})^{+}:=\mathbf{G}(\mathbb{Q}) \cap GG(Q)+:=G(Q)∩G. It is thus naturally a complex analytic variety, which is smooth if Γ Î“ Gamma\GammaΓ is torsion-free. The special subvarieties of Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D are the images of the special subvarieties of D D DDD under the projection π : D Γ D Ï€ : D → Γ ∖ D pi:D rarr Gamma\\D\pi: D \rightarrow \Gamma \backslash DÏ€:D→Γ∖D (one easily checks these are closed complex analytic subvarieties of Γ D ) Γ ∖ D ) Gamma\\D)\Gamma \backslash D)Γ∖D). For any algebraic representation λ : G G L ( V Q ) λ : G → G L V Q lambda:GrarrGL(V_(Q))\lambda: \mathbf{G} \rightarrow \mathbf{G L}\left(V_{\mathbb{Q}}\right)λ:G→GL(VQ), the G ( Q ) G ( Q ) G(Q)\mathbf{G}(\mathbb{Q})G(Q)-equivariant local system V ~ λ V ~ λ widetilde(V)_(lambda)\widetilde{V}_{\lambda}V~λ as well as the filtered holomorphic vector bundle ( V ˇ λ , F ) V ˇ λ , F ∙ (V^(ˇ)_(lambda),F^(∙))\left(\check{\mathcal{V}}_{\lambda}, F^{\bullet}\right)(Vˇλ,F∙) on D ˇ D ˇ D^(ˇ)\check{D}Dˇ are G G GGG-equivariant when restricted to D D DDD, hence descend to a triple ( V λ , ( V λ , F ) , ) V λ , V λ , F ∙ , ∇ (V_(lambda),(V_(lambda),F^(∙)),grad)\left(\mathbb{V}_{\lambda},\left(\mathcal{V}_{\lambda}, F^{\bullet}\right), \nabla\right)(Vλ,(Vλ,F∙),∇) on Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D. Similarly, the horizontal tangent bundle of D ˇ D ˇ D^(ˇ)\check{D}Dˇ defines the horizontal tangent bundle T h ( Γ D ) T ( Γ D ) T h ( Γ ∖ D ) ⊂ T ( Γ ∖ D ) T_(h)(Gamma\\D)sub T(Gamma\\D)T_{h}(\Gamma \backslash D) \subset T(\Gamma \backslash D)Th(Γ∖D)⊂T(Γ∖D) of the Hodge variety Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D.

2.5. Polarized Z Z Z\mathbb{Z}Z-variations of Hodge structures

Hodge theory as recalled in Section 2.1 can be considered as the particular case over a point of Hodge theory over an arbitrary base. Again, the motivation comes from geometry. Let f : Y B f : Y → B f:Y rarr Bf: Y \rightarrow Bf:Y→B be a proper surjective complex analytic submersion from a connected Kähler manifold Y Y YYY to a complex manifold B B BBB. It defines a locally constant sheaf V Z := R f Z V Z := R ∙ f ∗ Z V_(Z):=R^(∙)f_(**)Z\mathbb{V}_{\mathbb{Z}}:=R^{\bullet} f_{*} \mathbb{Z}VZ:=R∙f∗Z of finitely generated abelian groups on B B BBB, gathering the cohomologies H B ( Y b , Z ) , b B H B ∙ Y b , Z , b ∈ B H_(B)^(∙)(Y_(b),Z),b in BH_{\mathrm{B}}^{\bullet}\left(Y_{b}, \mathbb{Z}\right), b \in BHB∙(Yb,Z),b∈B. Upon choosing a base point b 0 B b 0 ∈ B b_(0)in Bb_{0} \in Bb0∈B, the datum of V Z V Z V_(Z)\mathbb{V}_{\mathbb{Z}}VZ is equivalent to the datum of a monodromy representation ρ : π 1 ( B , b 0 ) G L ( V Z , b 0 ) ρ : Ï€ 1 B , b 0 → G L V Z , b 0 rho:pi_(1)(B,b_(0))rarrGL(V_(Z,b_(0)))\rho: \pi_{1}\left(B, b_{0}\right) \rightarrow \mathbf{G L}\left(\mathbb{V}_{\mathbb{Z}, b_{0}}\right)ρ:Ï€1(B,b0)→GL(VZ,b0). On the other hand, the de Rham incarnation of the cohomology of the fibers of f f fff is the holomorphic flat vector bundle ( V := V Z Z B O B V := V Z ⊗ Z B O B ≃ (V:=V_(Z)ox_(Z_(B))O_(B)≃:}\left(\mathcal{V}:=\mathbb{V}_{\mathbb{Z}} \otimes_{\mathbb{Z}_{B}} \mathcal{O}_{B} \simeq\right.(V:=VZ⊗ZBOB≃ R f Ω Y / B , ) R ∙ f ∗ Ω Y / B ∙ , ∇ {:R^(∙)f_(**)Omega_(Y//B)^(∙),grad)\left.R^{\bullet} f_{*} \Omega_{Y / B}^{\bullet}, \nabla\right)R∙f∗ΩY/B∙,∇), where O B O B O_(B)\mathcal{O}_{B}OB is the sheaf of holomorphic functions on B , Ω Y / B B , Ω Y / B ∙ B,Omega_(Y//B)^(∙)B, \Omega_{Y / B}^{\bullet}B,ΩY/B∙ is the relative holomorphic de Rham complex and ∇ grad\nabla∇ is the Gauss-Manin connection. The Hodge filtration on each H B ( Y b , C ) H B ∙ Y b , C H_(B)^(∙)(Y_(b),C)H_{\mathrm{B}}^{\bullet}\left(Y_{b}, \mathbb{C}\right)HB∙(Yb,C) is induced by the holomorphic subbundles F p := R f Ω Y / B p F p := R ∙ f ∗ Ω Y / B ∙ ≥ p F^(p):=R^(∙)f_(**)Omega_(Y//B)^(∙ >= p)F^{p}:=R^{\bullet} f_{*} \Omega_{Y / B}^{\bullet \geq p}Fp:=R∙f∗ΩY/B∙≥p of V V V\mathcal{V}V. The Hodge filtration is usually not preserved by the connection, but Griffiths [42] crucially observed that it satisfies the transversality constraint F p Ω B 1 O B F p 1 ∇ F p ⊂ Ω B 1 ⊗ O B F p − 1 gradF^(p)subOmega_(B)^(1)ox_(O_(B))F^(p-1)\nabla F^{p} \subset \Omega_{B}^{1} \otimes_{\mathcal{O}_{B}} F^{p-1}∇Fp⊂ΩB1⊗OBFp−1. More generally, a variation of Z Z Z\mathbb{Z}Z-Hodge structures ( Z V H S Z V H S ZVHS\mathbb{Z} V H SZVHS ) on a connected complex manifold ( B , O B ) B , O B (B,O_(B))\left(B, \mathcal{O}_{B}\right)(B,OB) is a pair V := ( V Z , F ) V := V Z , F ∙ V:=(V_(Z),F^(∙))\mathbb{V}:=\left(\mathbb{V}_{\mathbb{Z}}, F^{\bullet}\right)V:=(VZ,F∙), consisting of a locally constant sheaf of finitely gener
ated abelian groups V Z V Z V_(Z)\mathbb{V}_{\mathbb{Z}}VZ on B B BBB and a (decreasing) filtration F F ∙ F^(∙)F^{\bullet}F∙ of the holomorphic vector bundle V := V Z Z B O B V := V Z ⊗ Z B O B V:=V_(Z)ox_(Z_(B))O_(B)\mathcal{V}:=\mathbb{V}_{\mathbb{Z}} \otimes_{\mathbb{Z}_{B}} \mathcal{O}_{B}V:=VZ⊗ZBOB by holomorphic subbundles, called the Hodge filtration, satisfying the following conditions: for each b B b ∈ B b in Bb \in Bb∈B, the pair ( V b , F b ) V b , F b ∙ (V_(b),F_(b)^(∙))\left(\mathbb{V}_{b}, F_{b}^{\bullet}\right)(Vb,Fb∙) is a Z Z Z\mathbb{Z}Z-Hodge structure; and the flat connection ∇ grad\nabla∇ on V V V\mathcal{V}V defined by V C V C V_(C)\mathbb{V}_{\mathbb{C}}VC satisfies Griffiths' transversality,
(2.1) F Ω B 1 O B F 1 (2.1) ∇ F ∙ ⊂ Ω B 1 ⊗ O B F ∙ − 1 {:(2.1)gradF^(∙)subOmega_(B)^(1)ox_(O_(B))F^(∙-1):}\begin{equation*} \nabla F^{\bullet} \subset \Omega_{B}^{1} \otimes_{\mathcal{O}_{B}} F^{\bullet-1} \tag{2.1} \end{equation*}(2.1)∇F∙⊂ΩB1⊗OBF∙−1
A morphism V V V → V ′ VrarrV^(')\mathbb{V} \rightarrow \mathbb{V}^{\prime}V→V′ of Z V H S s Z V H S s ZVHSs\mathbb{Z V H S s}ZVHSs on B B BBB is a morphism f : V Z V Z f : V Z → V Z ′ f:V_(Z)rarrV_(Z)^(')f: \mathbb{V}_{\mathbb{Z}} \rightarrow \mathbb{V}_{\mathbb{Z}}^{\prime}f:VZ→VZ′ of local systems such that the associated morphism of vector bundles f : V V f : V → V ′ f:VrarrV^(')f: \mathcal{V} \rightarrow \mathcal{V}^{\prime}f:V→V′ is compatible with the Hodge filtrations. If V V V\mathbb{V}V has weight k k kkk, a polarization of V V V\mathbb{V}V is a morphism q : V V Z B ( k ) q : V ⊗ V → Z B ( − k ) q:VoxVrarrZ_(B)(-k)\mathrm{q}: \mathbb{V} \otimes \mathbb{V} \rightarrow \mathbb{Z}_{B}(-k)q:V⊗V→ZB(−k) inducing a polarization on each Z Z Z\mathbb{Z}Z-Hodge structure V b , b B V b , b ∈ B V_(b),b in B\mathbb{V}_{b}, b \in BVb,b∈B. In the geometric situation, such a polarization exists if there exists an element η H 2 ( Y , Z ) η ∈ H 2 ( Y , Z ) eta inH^(2)(Y,Z)\eta \in H^{2}(Y, \mathbb{Z})η∈H2(Y,Z) whose restriction to each fiber Y b Y b Y_(b)Y_{b}Yb defines a Kähler class, for instance if f f fff is the analytification of a smooth projective morphism of smooth connected algebraic varieties over C C C\mathbb{C}C.

2.6. Generic Hodge datum and period map

Let S S SSS be a smooth connected quasiprojective variety over C C C\mathbb{C}C and let V V V\mathbb{V}V be a polarized Z V H S Z V H S ZVHS\mathbb{Z V H S}ZVHS on S an S an  S^("an ")S^{\text {an }}San . Fix a base point o S an o ∈ S an  o inS^("an ")o \in S^{\text {an }}o∈San , let p : S an ~ S an p : S an  ~ → S an  p: widetilde(S^("an "))rarrS^("an ")p: \widetilde{S^{\text {an }}} \rightarrow S^{\text {an }}p:San ~→San  be the corresponding universal cover and write V Z := V Z , o , q Z := q Z , o V Z := V Z , o , q Z := q Z , o V_(Z):=V_(Z,o),q_(Z):=q_(Z,o)V_{\mathbb{Z}}:=\mathbb{V}_{\mathbb{Z}, o}, q_{\mathbb{Z}}:=\mathrm{q}_{\mathbb{Z}, o}VZ:=VZ,o,qZ:=qZ,o. The pulled-back polarized Z V H S p V Z V H S p ∗ V ZVHSp^(**)V\mathbb{Z} V H S p^{*} \mathbb{V}ZVHSp∗V is canonically trivialized as ( S a n ~ × V Z , ( S an ~ × V C , F ) , = d , q Z ) S a n ~ × V Z , S an  ~ × V C , F ∙ , ∇ = d , q Z (( widetilde(S^(an)))xxV_(Z),(( widetilde(S^("an ")))xxV_(C),F^(∙)),grad=d,q_(Z))\left(\widetilde{S^{\mathrm{an}}} \times V_{\mathbb{Z}},\left(\widetilde{S^{\text {an }}} \times V_{\mathbb{C}}, F^{\bullet}\right), \nabla=d, q_{\mathbb{Z}}\right)(San~×VZ,(San ~×VC,F∙),∇=d,qZ). In [31, 7.5], Deligne proved that there exists a reductive Q Q Q\mathbb{Q}Q-algebraic subgroup ι : G G L ( V Q ) ι : G ↪ G L V Q iota:G↪GL(V_(Q))\iota: \mathbf{G} \hookrightarrow \mathbf{G L}\left(V_{\mathbb{Q}}\right)ι:G↪GL(VQ), called the generic Mumford-

contained in G G G\mathbf{G}G, and is equal to G G G\mathbf{G}G outside of a meagre set of S an ~ S an  ~ widetilde(S^("an "))\widetilde{S^{\text {an }}}San ~ (such points s ~ s ~ tilde(s)\tilde{s}s~ are said Hodge generic for V V V\mathbb{V}V ). A closed irreducible subvariety Y S Y ⊂ S Y sub SY \subset SY⊂S is said Hodge generic for V V V\mathbb{V}V if it contains a Hodge generic point. The setup of Section 2.3 is thus in force. Without loss of generality, we can assume that the point o ~ o ~ tilde(o)\tilde{o}o~ is Hodge generic. Let ( G , D ) ( G , D ) (G,D)(\mathbf{G}, D)(G,D) be the Hodge datum (called the generic Hodge datum of S an S an  S^("an ")S^{\text {an }}San  for V V V\mathbb{V}V ) associated with the polarized Hodge structure ( V Z , F o ~ ) V Z , F o ~ ∙ (V_(Z),F_( tilde(o))^(∙))\left(V_{\mathbb{Z}}, F_{\tilde{o}}^{\bullet}\right)(VZ,Fo~∙). The Z V H S p V Z V H S p ∗ V ZVHSp^(**)V\mathbb{Z} V H S p^{*} \mathbb{V}ZVHSp∗V is completely described by a holomorphic map Φ ~ : S a n ~ D Φ ~ : S a n ~ → D widetilde(Phi): widetilde(S^(an))rarr D\widetilde{\Phi}: \widetilde{S^{\mathrm{an}}} \rightarrow DΦ~:San~→D, which is naturally equivariant under the monodromy representation ρ : π 1 ( S an , o ) Γ := G G L ( V Z ) ρ : Ï€ 1 S an  , o → Γ := G ∩ G L V Z rho:pi_(1)(S^("an "),o)rarr Gamma:=G nnGL(V_(Z))\rho: \pi_{1}\left(S^{\text {an }}, o\right) \rightarrow \Gamma:=G \cap \mathbf{G L}\left(V_{\mathbb{Z}}\right)ρ:Ï€1(San ,o)→Γ:=G∩GL(VZ), hence descends to a holomorphic map Φ : S an Γ D Φ : S an  → Γ ∖ D Phi:S^("an ")rarr Gamma\\D\Phi: S^{\text {an }} \rightarrow \Gamma \backslash DΦ:San →Γ∖D, called the period map of S S SSS for V V V\mathbb{V}V. We thus obtain the following commutative diagram in the category of complex analytic spaces:
Notice that the pair ( V Q , ( V , F ) ) V Q , V , F ∙ (V_(Q),(V,F^(∙)))\left(\mathbb{V}_{\mathbb{Q}},\left(\mathcal{V}, F^{\bullet}\right)\right)(VQ,(V,F∙)) is the pullback under Φ Î¦ Phi\PhiΦ of the pair ( V ι , ( V ι , F ) ) V ι , V ι , F ∙ (V_(iota),(V_(iota),F^(∙)))\left(\mathbb{V}_{\iota},\left(\mathcal{V}_{\iota}, F^{\bullet}\right)\right)(Vι,(Vι,F∙)) on the Hodge variety Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D defined by the inclusion ι : G G L ( V Q ) ι : G ↪ G L V Q iota:G↪GL(V_(Q))\iota: \mathbf{G} \hookrightarrow \mathbf{G L}\left(V_{\mathbb{Q}}\right)ι:G↪GL(VQ). Griffiths' transversality condition is equivalent to the statement that Φ Î¦ Phi\PhiΦ is horizontal, d Φ ( T S a n ) T h ( Γ D ) d Φ T S a n ⊂ T h ( Γ ∖ D ) d Phi(TS^(an))subT_(h)(Gamma\\D)d \Phi\left(T S^{\mathrm{an}}\right) \subset T_{h}(\Gamma \backslash D)dΦ(TSan)⊂Th(Γ∖D). By extension we call period map any holomorphic, horizontal, locally liftable map from S an S an  S^("an ")S^{\text {an }}San  to a Hodge variety Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D.
The Hodge locus H L ( S , V ) H L S , V ⊗ HL(S,V^(ox))\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)HL(S,V⊗) of S S SSS for V V V\mathbb{V}V is the subset of points s S an s ∈ S an  s inS^("an ")s \in S^{\text {an }}s∈San  for which the Mumford-Tate group G s G s G_(s)\mathbf{G}_{s}Gs is a strict subgroup of G G G\mathbf{G}G, or equivalently for which the Hodge structure V s V s V_(s)\mathbb{V}_{s}Vs admits more Hodge tensors than the very general fiber V s V s ′ V_(s^('))\mathbb{V}_{s^{\prime}}Vs′. Thus
(2.3) H L ( S , V ) = ( G , D ) ( G , D ) Φ 1 ( Γ D ) (2.3) H L S , V ⊗ = ⋃ G ′ , D ′ ↪ ( G , D )   Φ − 1 Γ ′ ∖ D ′ {:(2.3)HL(S,V^(ox))=uuu_((G^('),D^('))↪(G,D))Phi^(-1)(Gamma^(')\\D^(')):}\begin{equation*} \mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)=\bigcup_{\left(\mathbf{G}^{\prime}, D^{\prime}\right) \hookrightarrow(\mathbf{G}, D)} \Phi^{-1}\left(\Gamma^{\prime} \backslash D^{\prime}\right) \tag{2.3} \end{equation*}(2.3)HL(S,V⊗)=⋃(G′,D′)↪(G,D)Φ−1(Γ′∖D′)
where the union is over all strict Hodge subdata and Γ D Γ ′ ∖ D ′ Gamma^(')\\D^(')\Gamma^{\prime} \backslash D^{\prime}Γ′∖D′ is a slight abuse of notation for denoting the projection of D D D ′ ⊂ D D^(')sub DD^{\prime} \subset DD′⊂D to Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D.
Let Y S Y ⊂ S Y sub SY \subset SY⊂S be a closed irreducible algebraic subvariety i : Y S i : Y ↪ S i:Y↪Si: Y \hookrightarrow Si:Y↪S. Let ( G Y , D Y ) G Y , D Y (G_(Y),D_(Y))\left(\mathbf{G}_{Y}, D_{Y}\right)(GY,DY) be the generic Hodge datum of the Z V H S V Z V H S V ZVHSV\mathbb{Z} V H S \mathbb{V}ZVHSV restricted to the smooth locus of Y Y YYY. The algebraic monodromy group H Y H Y H_(Y)\mathbf{H}_{Y}HY of Y Y YYY for V V V\mathbb{V}V is the identity component of the Zariski-closure in G L ( V Q ) G L V Q GL(V_(Q))\mathbf{G L}\left(V_{\mathbb{Q}}\right)GL(VQ) of the monodromy of the restriction to Y Y YYY of the local system V Z V Z V_(Z)\mathbb{V}_{\mathbb{Z}}VZ. It follows from Deligne's (in the geometric case) and Schmid's (in general) "Theorem of the fixed part" and "Semisimplicity Theorem" that H Y H Y H_(Y)\mathbf{H}_{Y}HY is a normal subgroup of the derived group G Y d e r G Y d e r G_(Y)^(der)\mathbf{G}_{Y}^{\mathrm{der}}GYder, see [2, THEOREM 1].

3. HODGE THEORY AND TAME GEOMETRY

3.1. Variational Hodge theory between algebraicity and transcendence

Let S S SSS be a smooth connected quasi-projective variety over C C C\mathbb{C}C and let V = ( V Z , F ) V = V Z , F ∙ V=(V_(Z),F^(∙))\mathbb{V}=\left(\mathbb{V}_{\mathbb{Z}}, F^{\bullet}\right)V=(VZ,F∙) be a polarized Z V H S Z V H S ZVHS\mathbb{Z V H S}ZVHS on S an S an  S^("an ")S^{\text {an }}San . Let ( G , D ) ( G , D ) (G,D)(\mathbf{G}, D)(G,D) be the generic Hodge datum of S S SSS for V V V\mathbb{V}V and let Φ : S a n Γ D Φ : S a n → Γ ∖ D Phi:S^(an)rarr Gamma\\D\Phi: S^{\mathrm{an}} \rightarrow \Gamma \backslash DΦ:San→Γ∖D be the period map defined by V V V\mathbb{V}V.
The fact that Hodge theory is a transcendental theory is reflected in the following facts:
  • First, the triplets ( V λ , ( V λ , F ) , ) V λ , V λ , F ∙ , ∇ (V_(lambda),(V_(lambda),F^(∙)),grad)\left(\mathbb{V}_{\lambda},\left(\mathcal{V}_{\lambda}, F^{\bullet}\right), \nabla\right)(Vλ,(Vλ,F∙),∇) on Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D (for λ : G G L ( V Q ) λ : G → G L V Q lambda:GrarrGL(V_(Q))\lambda: \mathbf{G} \rightarrow \mathbf{G L}\left(V_{\mathbb{Q}}\right)λ:G→GL(VQ) an algebraic representation) do not in general satisfy Griffiths' transversality, hence do not define a Z V H S Z V H S ZVHS\mathbb{Z V H S}ZVHS on Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D. They do if and only if V V V\mathbb{V}V is of Shimura type, i.e., ( G , D ) ( G , D ) (G,D)(\mathbf{G}, D)(G,D) is a (connected) Shimura datum (meaning that the weight zero Hodge structures on the fibers of V Ad V Ad  V_("Ad ")\mathbb{V}_{\text {Ad }}VAd  are of type { ( 1 , 1 ) , ( 0 , 0 ) , ( 1 , 1 ) } ) { ( − 1 , 1 ) , ( 0 , 0 ) , ( 1 , − 1 ) } {:{(-1,1),(0,0),(1,-1)})\left.\{(-1,1),(0,0),(1,-1)\}\right){(−1,1),(0,0),(1,−1)}); or equivalently, if the horizontal tangent bundle T h D T h D T_(h)DT_{h} DThD coincides with T D T D TDT DTD. In other words, Hodge varieties are in general not classifying spaces for polarized Z V H S Z V H S ZVHS\mathbb{Z V H S}ZVHS.
  • Second, and more importantly, the complex analytic Hodge variety Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D is in general not algebraizable (i.e., it is not the analytification of a complex quasiprojective variety). More precisely, let us write D = G / M D = G / M D=G//MD=G / MD=G/M as in Section 2.3. A classical property of elliptic orbits like D D DDD is that there exists a unique maximal compact subgroup K K KKK of G G GGG containing M M MMM [46]. Supposing for simplicity that G G GGG is a real simple Lie group G G GGG, then Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D is algebraizable only if G / K G / K G//KG / KG/K is a hermitian symmetric domain and the projection D G / K D → G / K D rarr G//KD \rightarrow G / KD→G/K is holomorphic or antiholomorphic, see [ 45 ] [ 45 ] [45][45][45].
On the other hand, this transcendence is severely constrained, as shown by the following algebraicity results:
  • If ( G , D ) ( G , D ) (G,D)(\mathbf{G}, D)(G,D) is of Shimura type then Γ D = S h a n Γ ∖ D = S h a n Gamma\\D=Sh^(an)\Gamma \backslash D=\mathrm{Sh}^{\mathrm{an}}Γ∖D=Shan is the analytification of an algebraic variety, called a Shimura variety Sh [8,30,32]. In that case Borel [17, THEOREM 3.10] proved that the complex analytic period map Φ : S a n S h an Φ : S a n → S h an  Phi:S^(an)rarrSh^("an ")\Phi: S^{\mathrm{an}} \rightarrow \mathrm{Sh}^{\text {an }}Φ:San→Shan  is the analytification of an algebraic map.
  • Let S S ¯ S ⊂ S ¯ S sub bar(S)S \subset \bar{S}S⊂S¯ be a log-smooth compactification of S S SSS by a simple normal crossing divisor Z Z ZZZ. Following Deligne [28], the flat holomorphic connection ∇ grad\nabla∇ on V V V\mathcal{V}V defines a canonical extension V ¯ V ¯ bar(V)\overline{\mathcal{V}}V¯ of V V V\mathcal{V}V to S ¯ S ¯ bar(S)\bar{S}S¯. Using GAGA for S ¯ S ¯ bar(S)\bar{S}S¯, this defines an algebraic structure on ( V , ) ( V , ∇ ) (V,grad)(\mathcal{V}, \nabla)(V,∇), for which the connection ∇ grad\nabla∇ is regular. Around any point of Z Z ZZZ, the complex manifold S an S an  S^("an ")S^{\text {an }}San  is locally isomorphic to a product ( Δ ) k × Δ l Δ ∗ k × Δ l (Delta^(**))^(k)xxDelta^(l)\left(\Delta^{*}\right)^{k} \times \Delta^{l}(Δ∗)k×Δl of punctured polydisks. Borel showed that the monodromy representation ρ : π 1 ( S an , s o ) Γ G ( Q ) ρ : Ï€ 1 S an  , s o → Γ ⊂ G ( Q ) rho:pi_(1)(S^("an "),s_(o))rarr Gamma subG(Q)\rho: \pi_{1}\left(S^{\text {an }}, s_{o}\right) \rightarrow \Gamma \subset \mathbf{G}(\mathbb{Q})ρ:Ï€1(San ,so)→Γ⊂G(Q) of V V V\mathbb{V}V is "tame at infinity," that is, its restriction to Z k = π 1 ( ( Δ ) k × Δ l ) Z k = Ï€ 1 Δ ∗ k × Δ l Z^(k)=pi_(1)((Delta^(**))^(k)xxDelta^(l))\mathbb{Z}^{k}=\pi_{1}\left(\left(\Delta^{*}\right)^{k} \times \Delta^{l}\right)Zk=Ï€1((Δ∗)k×Δl) is quasiunipotent, see [82, (4.5)]. Using this result, Schmid showed that the Hodge filtration F F ∙ F^(∙)F^{\bullet}F∙ extends holomorphically to the Deligne extension V ¯ V ¯ bar(V)\overline{\mathcal{V}}V¯. This is the celebrated Nilpotent Orbit theorem [82, (4.12)]. It follows, as noticed by Griffiths [82, (4.13)], that the Hodge filtration on V V V\mathcal{V}V comes from an algebraic filtration on the underlying algebraic bundle, whether V V V\mathbb{V}V is of geometric origin or not.
  • More recently, an even stronger evidence came from the study of Hodge loci. Cattani, Deligne, and Kaplan proved the following celebrated result (generalized to the mixed case in [18-21]):
Theorem 3.1 ([22]). Let S S SSS be a smooth connected quasiprojective variety over C C C\mathbb{C}C and V V V\mathbb{V}V be a polarized Z V H S Z V H S ZVHS\mathbb{Z} V H SZVHS over S S SSS. Then H L ( S , V ) H L S , V ⊗ HL(S,V^(ox))\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)HL(S,V⊗) is a countable union of closed irreducible algebraic subvarieties of S S SSS.
In view of this tension between algebraicity and transcendence, it is natural to ask if there is a framework, less strict than complex algebraic geometry but more constraining than complex analytic geometry, where to analyze period maps and explain its remarkable properties.

3.2. O-minimal geometry

Such a framework was in fact envisioned by Grothendieck in [47, 85] under the name "tame topology," as a way out of the pathologies of general topological spaces. Examples of pathologies are Cantor sets, space-filling curves but also much simpler objects like the graph Γ := { ( x , sin 1 x ) , 0 < x 1 } R 2 : Γ := x , sin ⁡ 1 x , 0 < x ≤ 1 ⊂ R 2 : Gamma:={(x,sin ((1)/(x))),0 < x <= 1}subR^(2):\Gamma:=\left\{\left(x, \sin \frac{1}{x}\right), 0<x \leq 1\right\} \subset \mathbb{R}^{2}:Γ:={(x,sin⁡1x),0<x≤1}⊂R2: its closure Γ ¯ := Γ ⨿ I Γ ¯ := Γ ⨿ I bar(Gamma):=Gamma⨿I\bar{\Gamma}:=\Gamma \amalg \mathrm{I}Γ¯:=Γ⨿I, where I := { 0 } × [ 1 , 1 ] R 2 I := { 0 } × [ − 1 , 1 ] ⊂ R 2 I:={0}xx[-1,1]subR^(2)\mathrm{I}:=\{0\} \times[-1,1] \subset \mathbb{R}^{2}I:={0}×[−1,1]⊂R2 is connected but not arc-connected; dim ( Γ ¯ Γ ) = dim Γ dim ⁡ ( Γ ¯ ∖ Γ ) = dim ⁡ Γ dim( bar(Gamma)\\Gamma)=dim Gamma\operatorname{dim}(\bar{\Gamma} \backslash \Gamma)=\operatorname{dim} \Gammadim⁡(Γ¯∖Γ)=dim⁡Γ, which prevents any reasonable stratification theory; and Γ R Γ ∩ R Gamma nnR\Gamma \cap \mathbb{R}Γ∩R is not "of finite type." Tame geometry has been developed by model theorists as o-minimal geometry, which studies structures where every definable set has a finite geometric complexity. Its prototype is real semialgebraic geometry, but it is much richer. We refer to [34] for a nice survey.
Definition 3.2. A structure S S SSS expanding the real field is a collection S = ( S n ) n N S = S n n ∈ N S=(S_(n))_(n inN)S=\left(S_{n}\right)_{n \in \mathbb{N}}S=(Sn)n∈N, where S n S n S_(n)S_{n}Sn is a set of subsets of R n R n R^(n)\mathbb{R}^{n}Rn such that for every n N n ∈ N n inNn \in \mathbb{N}n∈N :
(1) all algebraic subsets of R n R n R^(n)\mathbb{R}^{n}Rn are in S n S n S_(n)S_{n}Sn.
(2) S n S n S_(n)S_{n}Sn is a boolean subalgebra of the power set of R n R n R^(n)\mathbb{R}^{n}Rn (i.e., S n S n S_(n)S_{n}Sn is stable by finite union, intersection, and complement).
(3) If A S n A ∈ S n A inS_(n)A \in S_{n}A∈Sn and B S m B ∈ S m B inS_(m)B \in S_{m}B∈Sm then A × B S n + m A × B ∈ S n + m A xx B inS_(n+m)A \times B \in S_{n+m}A×B∈Sn+m.
(4) Let p : R n + 1 R n p : R n + 1 → R n p:R^(n+1)rarrR^(n)p: \mathbb{R}^{n+1} \rightarrow \mathbb{R}^{n}p:Rn+1→Rn be a linear projection. If A S n + 1 A ∈ S n + 1 A inS_(n+1)A \in S_{n+1}A∈Sn+1 then p ( A ) S n p ( A ) ∈ S n p(A)inS_(n)p(A) \in S_{n}p(A)∈Sn.
The elements of S n S n S_(n)S_{n}Sn are called the S S S\mathcal{S}S-definable sets of R n R n R^(n)\mathbb{R}^{n}Rn. A map f : A B f : A → B f:A rarr Bf: A \rightarrow Bf:A→B between S S S\mathcal{S}S definable sets is said to be S S S\mathcal{S}S-definable if its graph is S S S\mathcal{S}S-definable.
A dual point of view starts from the functions, namely considers sets definable in a first-order structure S = R , + , × , < , ( f i ) i I S = R , + , × , < , f i i ∈ I S=(:R,+,xx, < ,(f_(i))_(i in I):)S=\left\langle\mathbb{R},+, \times,<,\left(f_{i}\right)_{i \in I}\right\rangleS=⟨R,+,×,<,(fi)i∈I⟩ where I I III is a set and the f i : R n i R , i I f i : R n i → R , i ∈ I f_(i):R^(n_(i))rarrR,i in If_{i}: \mathbb{R}^{n_{i}} \rightarrow \mathbb{R}, i \in Ifi:Rni→R,i∈I, are functions. A subset Z R n Z ⊂ R n Z subR^(n)Z \subset \mathbb{R}^{n}Z⊂Rn is S S SSS-definable if it can be defined by a formula
Z := { ( x 1 , , x n ) R n ϕ ( x 1 , , x n ) is true } Z := x 1 , … , x n ∈ R n ∣ Ï• x 1 , … , x n  is true  Z:={(x_(1),dots,x_(n))inR^(n)∣phi(x_(1),dots,x_(n))" is true "}Z:=\left\{\left(x_{1}, \ldots, x_{n}\right) \in \mathbb{R}^{n} \mid \phi\left(x_{1}, \ldots, x_{n}\right) \text { is true }\right\}Z:={(x1,…,xn)∈Rn∣ϕ(x1,…,xn) is true }
where ϕ Ï• phi\phiÏ• is a first-order formula that can be written using only the quantifiers ∀ AA\forall∀ and ∃ EE\exists∃ applied to real variables; logical connectors; algebraic expressions written with the f i f i f_(i)f_{i}fi; the order symbol < < <<<; and fixed parameters λ i R λ i ∈ R lambda_(i)inR\lambda_{i} \in \mathbb{R}λi∈R. When the set I I III is empty the S S SSS-definable subsets are the semialgebraic sets. Semialgebraic subsets are thus always S S SSS-definable.
One easily checks that the composite of δ δ delta\deltaδ-definable functions is δ δ delta\deltaδ-definable, as are the images and the preimages of δ δ delta\deltaδ-definable sets under δ δ delta\deltaδ-definable maps. Using that the euclidean distance is a real-algebraic function, one shows easily that the closure and interior of an S S S\mathcal{S}S-definable set are again S S S\mathcal{S}S-definable.
The following o-minimal axiom for a structure S S SSS guarantees the possibility of doing geometry using S S S\mathcal{S}S-definable sets as basic blocks.
Definition 3.3. A structure S S SSS is said to be o-minimal if S 1 S 1 S_(1)S_{1}S1 consists precisely of the finite unions of points and intervals (i.e., the semialgebraic subsets of R R R\mathbb{R}R ).
Example 3.4. The structure R sin := R , + , × , < , sin R sin := ⟨ R , + , × , < , sin ⟩ R_(sin):=(:R,+,xx, < ,sin:)\mathbb{R}_{\sin }:=\langle\mathbb{R},+, \times,<, \sin \rangleRsin:=⟨R,+,×,<,sin⟩ is not o-minimal. Indeed, the infinite union of points π Z = { x R sin x = 0 } Ï€ Z = { x ∈ R ∣ sin ⁡ x = 0 } piZ={x inR∣sin x=0}\pi \mathbb{Z}=\{x \in \mathbb{R} \mid \sin x=0\}Ï€Z={x∈R∣sin⁡x=0} is a definable subset of R R R\mathbb{R}R in this structure.
Any o-minimal structure S S SSS has the following main tameness property: given finitely many S S SSS-definable sets U 1 , , U k R n U 1 , … , U k ⊂ R n U_(1),dots,U_(k)subR^(n)U_{1}, \ldots, U_{k} \subset \mathbb{R}^{n}U1,…,Uk⊂Rn, there exists a definable cylindrical cellular decomposition of R n R n R^(n)\mathbb{R}^{n}Rn such that each U i U i U_(i)U_{i}Ui is a finite union of cells. Such a decomposition is defined inductively on n n nnn. For n = 1 n = 1 n=1n=1n=1, this is a finite partition of R R R\mathbb{R}R into cells which are points or open intervals. For n > 1 n > 1 n > 1n>1n>1, it is obtained from a definable cylindrical cellular decomposition of R n 1 R n − 1 R^(n-1)\mathbb{R}^{n-1}Rn−1 by fixing, for any cell C R n 1 C ⊂ R n − 1 C subR^(n-1)C \subset \mathbb{R}^{n-1}C⊂Rn−1, finitely many definable functions f C , i : C R f C , i : C → R f_(C,i):C rarrRf_{C, i}: C \rightarrow \mathbb{R}fC,i:C→R, 1 i k C 1 ≤ i ≤ k C 1 <= i <= k_(C)1 \leq i \leq k_{C}1≤i≤kC, with f C , 0 := < f C , 1 < < f C , k C < f C , k C + 1 := + f C , 0 := − ∞ < f C , 1 < ⋯ < f C , k C < f C , k C + 1 := + ∞ f_(C,0):=-oo < f_(C,1) < cdots < f_(C,k_(C)) < f_(C,k_(C)+1):=+oof_{C, 0}:=-\infty<f_{C, 1}<\cdots<f_{C, k_{C}}<f_{C, k_{C}+1}:=+\inftyfC,0:=−∞<fC,1<⋯<fC,kC<fC,kC+1:=+∞, and defining the cells of R n R n R^(n)\mathbb{R}^{n}Rn as the graphs { ( x , f C , i ( x ) ) , x C } , 1 i k C x , f C , i ( x ) , x ∈ C , 1 ≤ i ≤ k C {(x,f_(C,i)(x)),x in C},1 <= i <= k_(C)\left\{\left(x, f_{C, i}(x)\right), x \in C\right\}, 1 \leq i \leq k_{C}{(x,fC,i(x)),x∈C},1≤i≤kC, and the bands { ( x , f C , i ( x ) < x , f C , i ( x ) < {(x,f_(C,i)(x) < :}\left\{\left(x, f_{C, i}(x)<\right.\right.{(x,fC,i(x)< y < f C , i + 1 ( x ) ) , x C , y R } , 0 i k C y < f C , i + 1 ( x ) , x ∈ C , y ∈ R , 0 ≤ i ≤ k C {:y < f_(C,i+1)(x)),x in C,y inR},0 <= i <= k_(C)\left.\left.y<f_{C, i+1}(x)\right), x \in C, y \in \mathbb{R}\right\}, 0 \leq i \leq k_{C}y<fC,i+1(x)),x∈C,y∈R},0≤i≤kC, for all cells C C CCC of R n 1 R n − 1 R^(n-1)\mathbb{R}^{n-1}Rn−1.
The simplest o-minimal structure is the structure R alg R alg  R_("alg ")\mathbb{R}_{\text {alg }}Ralg  consisting of semialgebraic sets. It is too close to algebraic geometry to be used for studying transcendence phenomena. Luckily much richer o-minimal geometries do exist. A fundamental result of Wilkie, building on the result of Khovanskii [54] that any exponential set { ( x 1 , , x n ) x 1 , … , x n ∈ {(x_(1),dots,x_(n))in:}\left\{\left(x_{1}, \ldots, x_{n}\right) \in\right.{(x1,…,xn)∈ R n P ( x 1 , , x n , exp ( x 1 ) , , exp ( x n ) ) = 0 } ( R n ∣ P x 1 , … , x n , exp ⁡ x 1 , … , exp ⁡ x n = 0 {:R^(n)∣P(x_(1),dots,x_(n),exp(x_(1)),dots,exp(x_(n)))=0}(:}\left.\mathbb{R}^{n} \mid P\left(x_{1}, \ldots, x_{n}, \exp \left(x_{1}\right), \ldots, \exp \left(x_{n}\right)\right)=0\right\}\left(\right.Rn∣P(x1,…,xn,exp⁡(x1),…,exp⁡(xn))=0}( where P R [ X 1 , , X n , Y 1 , , Y n ] ) P ∈ R X 1 , … , X n , Y 1 , … , Y n {:P inR[X_(1),dots,X_(n),Y_(1),dots,Y_(n)])\left.P \in \mathbb{R}\left[X_{1}, \ldots, X_{n}, Y_{1}, \ldots, Y_{n}\right]\right)P∈R[X1,…,Xn,Y1,…,Yn]) has finitely many connected components, states:
Theorem 3.5 ([97]). The structure R exp := R , + , × , < , exp : R R R exp := ⟨ R , + , × , < , exp : R → R ⟩ R_(exp):=(:R,+,xx, < ,exp:RrarrR:)\mathbb{R}_{\exp }:=\langle\mathbb{R},+, \times,<, \exp : \mathbb{R} \rightarrow \mathbb{R}\rangleRexp:=⟨R,+,×,<,exp:R→R⟩ is o-minimal.
In another direction, let us define
R a n := R , + , × , < , { f } for f restricted real analytic function R a n := ⟨ R , + , × , < , { f }  for  f  restricted real analytic function  ⟩ R_(an):=(:R,+,xx, < ,{f}" for "f" restricted real analytic function ":)\mathbb{R}_{\mathrm{an}}:=\langle\mathbb{R},+, \times,<,\{f\} \text { for } f \text { restricted real analytic function }\rangleRan:=⟨R,+,×,<,{f} for f restricted real analytic function ⟩
where a function f : R n R f : R n → R f:R^(n)rarrRf: \mathbb{R}^{n} \rightarrow \mathbb{R}f:Rn→R is a restricted real analytic function if it is zero outside [ 0 , 1 ] n [ 0 , 1 ] n [0,1]^(n)[0,1]^{n}[0,1]n and if there exists a real analytic function g g ggg on a neighborhood of [ 0 , 1 ] n [ 0 , 1 ] n [0,1]^(n)[0,1]^{n}[0,1]n such that f f fff and g g ggg are equal on [ 0 , 1 ] n [ 0 , 1 ] n [0,1]^(n)[0,1]^{n}[0,1]n. Gabrielov's result [37] that the difference of two subanalytic sets is subanalytic implies rather easily that the structure R a n R a n R_(an)\mathbb{R}_{\mathrm{an}}Ran is o-minimal. The structure generated by two o-minimal structures is not o-minimal in general, but Van den Dries and Miller [35] proved that the structure R an,exp R an,exp  R_("an,exp ")\mathbb{R}_{\text {an,exp }}Ran,exp  generated by R an R an  R_("an ")\mathbb{R}_{\text {an }}Ran  and R exp R exp  R_("exp ")\mathbb{R}_{\text {exp }}Rexp  is o-minimal. This is the ominimal structure which will be mainly used in the rest of this text.
Let us now globalize the notion of definable set using charts:
Definition 3.6. A definable topological space X X XXX is the data of a Hausdorff topological space X X X\mathcal{X}X, a finite open covering ( U i ) 1 i k U i 1 ≤ i ≤ k (U_(i))_(1 <= i <= k)\left(U_{i}\right)_{1 \leq i \leq k}(Ui)1≤i≤k of X X X\mathcal{X}X, and homeomorphisms ψ i : U i V i R n ψ i : U i → V i ⊂ R n psi_(i):U_(i)rarrV_(i)subR^(n)\psi_{i}: U_{i} \rightarrow V_{i} \subset \mathbb{R}^{n}ψi:Ui→Vi⊂Rn such that all V i , V i j := ψ i ( U i U j ) V i , V i j := ψ i U i ∩ U j V_(i),V_(ij):=psi_(i)(U_(i)nnU_(j))V_{i}, V_{i j}:=\psi_{i}\left(U_{i} \cap U_{j}\right)Vi,Vij:=ψi(Ui∩Uj) and ψ i ψ j 1 : V i j V j i ψ i ∘ ψ j − 1 : V i j → V j i psi_(i)@psi_(j)^(-1):V_(ij)rarrV_(ji)\psi_{i} \circ \psi_{j}^{-1}: V_{i j} \rightarrow V_{j i}ψi∘ψj−1:Vij→Vji are definable. As usual the pairs ( U i , ψ i ) U i , ψ i (U_(i),psi_(i))\left(U_{i}, \psi_{i}\right)(Ui,ψi) are called charts. A morphism of definable topological spaces is a continuous map which is definable when read in the charts. The definable site X _ X X _ X X__(X)\underline{X}_{\mathcal{X}}X_X of a definable topological space X X X\mathcal{X}X has for objects definable open subsets U X U ⊂ X U sub XU \subset XU⊂X and admissible coverings are the finite ones.
Example 3.7. Let X X XXX be an algebraic variety over R R R\mathbb{R}R. Then X ( R ) X ( R ) X(R)X(\mathbb{R})X(R) equipped with the euclidean topology carries a natural R alg R alg  R_("alg ")\mathbb{R}_{\text {alg }}Ralg -definable structure (up to isomorphism): one covers X X XXX by finitely many (Zariski) open affine subvarieties X i X i X_(i)X_{i}Xi and take U i := X i ( R ) U i := X i ( R ) U_(i):=X_(i)(R)U_{i}:=X_{i}(\mathbb{R})Ui:=Xi(R) which is naturally a semialgebraic set. One easily check that any two finite open affine covers define isomorphic R alg R alg  R_("alg ")\mathbb{R}_{\text {alg }}Ralg -structures on X ( R ) X ( R ) X(R)X(\mathbb{R})X(R). If X X XXX is an algebraic variety over C C C\mathbb{C}C then X ( C ) = ( Res C / R X ) ( R ) X ( C ) = Res C ⁡ / R X ( R ) X(C)=(Res_(C)//RX)(R)X(\mathbb{C})=\left(\operatorname{Res}_{\mathbb{C}} / \mathbb{R} X\right)(\mathbb{R})X(C)=(ResC⁡/RX)(R) carries thus a natural R alg R alg  R_("alg ")\mathbb{R}_{\text {alg }}Ralg -structure. We call this the R alg R alg  R_("alg ")\mathbb{R}_{\text {alg }}Ralg -definabilization of X X XXX and denote it by X R alg X R alg  X^(R_("alg "))X^{\mathbb{R}_{\text {alg }}}XRalg .
In the rest of this section, we fix an o-minimal structure S S SSS and write "definable" for S S S\mathcal{S}S-definable. Given a complex algebraic variety X X XXX we write X def X def  X^("def ")X^{\text {def }}Xdef  for the S S S\mathcal{S}S-definabilization X S X S X^(S)X^{\mathcal{S}}XS.

3.3. O-minimal geometry and algebraization

Why should an algebraic geometer care about o-minimal geometry? Because ominimal geometry provides strong algebraization results.

3.3.1. Diophantine criterion

The first algebraization result is the celebrated Pila-Wilkie theorem:
Theorem 3.8 ([77]). Let Z R n Z ⊂ R n Z subR^(n)Z \subset \mathbb{R}^{n}Z⊂Rn be a definable set. We define Z alg Z alg  Z^("alg ")Z^{\text {alg }}Zalg  as the union of all connected positive-dimensional semialgebraic subsets of Z Z ZZZ. Then, denoting by H : Q n R H : Q n → R H:Q^(n)rarrRH: \mathbb{Q}^{n} \rightarrow \mathbb{R}H:Qn→R the standard height function:
ε > 0 , C ε > 0 , T > 0 , | { x ( Z Z a l g ) Q n , H ( x ) T } | < C ε T ε ∀ ε > 0 , ∃ C ε > 0 , ∀ T > 0 , x ∈ Z ∖ Z a l g ∩ Q n , H ( x ) ≤ T < C ε T ε AA epsi > 0,quad EEC_(epsi) > 0,quad AA T > 0,quad|{x in(Z\\Z^(alg))nnQ^(n),H(x) <= T}| < C_(epsi)T^(epsi)\forall \varepsilon>0, \quad \exists C_{\varepsilon}>0, \quad \forall T>0, \quad\left|\left\{x \in\left(Z \backslash Z^{\mathrm{alg}}\right) \cap \mathbb{Q}^{n}, H(x) \leq T\right\}\right|<C_{\varepsilon} T^{\varepsilon}∀ε>0,∃Cε>0,∀T>0,|{x∈(Z∖Zalg)∩Qn,H(x)≤T}|<CεTε
In words, if a definable set contains at least polynomially many rational points (with respect to their height), then it contains a positive dimensional semialgebraic set! For instance, if f : R R f : R → R f:RrarrRf: \mathbb{R} \rightarrow \mathbb{R}f:R→R is a real analytic function such that its graph Γ f [ 0 , 1 ] × [ 0 , 1 ] Γ f ∩ [ 0 , 1 ] × [ 0 , 1 ] Gamma_(f)nn[0,1]xx[0,1]\Gamma_{f} \cap[0,1] \times[0,1]Γf∩[0,1]×[0,1] contains at least polynomially many rational points (with respect to their height), then the function f f fff is real algebraic [15]. This algebraization result is a crucial ingredient in the proof of functional transcendence results for period maps, see Section 4.

3.3.2. Definable Chow and definable GAGA

In another direction, algebraicity follows from the meeting of o-minimal geometry with complex geometry. The motto is that o-minimal geometry is incompatible with the many pathologies of complex analysis. As a simple illustration, let f : Δ C f : Δ ∗ → C f:Delta^(**)rarrCf: \Delta^{*} \rightarrow \mathbb{C}f:Δ∗→C be a holomorphic function, and assume that f f fff is definable (where we identify C C C\mathbb{C}C with R 2 R 2 R^(2)\mathbb{R}^{2}R2 and Δ R 2 Δ ∗ ⊂ R 2 Delta^(**)subR^(2)\Delta^{*} \subset \mathbb{R}^{2}Δ∗⊂R2 is semialgebraic). Then f f fff does not have any essential singularity at 0 (i.e., f f fff is meromorphic). Otherwise, by the Big Picard theorem, the boundary Γ f ¯ Γ f Γ f ¯ ∖ Γ f bar(Gamma_(f))\\Gamma_(f)\overline{\Gamma_{f}} \backslash \Gamma_{f}Γf¯∖Γf of its graph would contain { 0 } × C { 0 } × C {0}xxC\{0\} \times \mathbb{C}{0}×C, hence would have the same real dimension (two) as Γ f Γ f Gamma_(f)\Gamma_{f}Γf, contradicting the fact that Γ f Γ f Gamma_(f)\Gamma_{f}Γf is definable.
Let us first define a good notion of a definable topological space "endowed with a complex analytic structure." We identify C n C n C^(n)\mathbb{C}^{n}Cn with R 2 n R 2 n R^(2n)\mathbb{R}^{2 n}R2n by taking real and imaginary parts. Given U C n U ⊂ C n U subC^(n)U \subset \mathbb{C}^{n}U⊂Cn a definable open subset, let O C n ( U ) O C n ( U ) O_(C^(n))(U)\mathcal{O}_{\mathbb{C}^{n}}(U)OCn(U) denote the C C C\mathbb{C}C-algebra of holomorphic definable functions U C U → C U rarrCU \rightarrow \mathbb{C}U→C. The assignment U O C n ( U ) U ⇝ O C n ( U ) U⇝O_(C^(n))(U)U \rightsquigarrow \mathcal{O}_{\mathbb{C}^{n}}(U)U⇝OCn(U) defines a sheaf O C n O C n O_(C^(n))\mathcal{O}_{\mathbb{C}^{n}}OCn on C n C n C^(n)\mathbb{C}^{n}Cn whose stalks are local rings. Given a finitely generated ideal I O C n ( U ) I ⊂ O C n ( U ) I subO_(C^(n))(U)I \subset \mathcal{O}_{\mathbb{C}^{n}}(U)I⊂OCn(U), its zero locus V ( I ) U V ( I ) ⊂ U V(I)sub UV(I) \subset UV(I)⊂U is definable and the restriction O V ( I ) := ( O U / I O U ) V ( I ) _ O V ( I ) := O U / I O U V ( I ) _ O_(V(I)):=(O_(U)//IO_(U))_(V(I)_)\mathcal{O}_{V(I)}:=\left(\mathcal{O}_{U} / I \mathcal{O}_{U}\right)_{\underline{V(I)}}OV(I):=(OU/IOU)V(I)_ define a sheaf of local rings on V ( I ) _ V ( I ) _ V(I)_\underline{V(I)}V(I)_.
Definition 3.9. A definable complex analytic space is a pair ( X , O X ) X , O X (X,O_(X))\left(\mathcal{X}, \mathcal{O}_{X}\right)(X,OX) consisting of a definable topological space X X X\mathcal{X}X and a sheaf O X O X O_(X)\mathcal{O}_{X}OX on X _ X _ X_\underline{X}X_ such that there exists a finite covering of X X X\mathcal{X}X by definable open subsets X i X i X_(i)\mathcal{X}_{i}Xi on which ( X , O X ) X i X , O X ∣ X i (X,O_(X))_(∣X_(i))\left(\mathcal{X}, \mathcal{O}_{X}\right){ }_{\mid X_{i}}(X,OX)∣Xi is isomorphic to some ( V ( I ) , O V ( I ) ) V ( I ) , O V ( I ) (V(I),O_(V(I)))\left(V(I), \mathcal{O}_{V(I)}\right)(V(I),OV(I)).
Bakker et al. [10, THEOREM 2.16] show that this is a reasonable definition: the sheaf O X O X O_(X)\mathcal{O}_{X}OX, in analogy with the classical Oka's theorem, is a coherent sheaf of rings. Moreover, one has a natural definabilization functor ( X , O X ) ( X def , O X def ) X , O X ⇝ X def  , O X def  (X,O_(X))⇝(X^("def "),O_(X^("def ")))\left(X, \mathcal{O}_{X}\right) \rightsquigarrow\left(X^{\text {def }}, \mathcal{O}_{X^{\text {def }}}\right)(X,OX)⇝(Xdef ,OXdef ) from the category of separated schemes (or algebraic spaces) of finite type over C C C\mathbb{C}C to the category of definable complex analytic spaces, which induces a morphism g : ( X def _ , O X def ) ( X _ , O X ) g : X def  _ , O X def  → X _ , O X g:(X^("def ")_,O_(X^("def ")))rarr(X_,O_(X))g:\left(\underline{X^{\text {def }}}, \mathcal{O}_{X^{\text {def }}}\right) \rightarrow\left(\underline{X}, \mathcal{O}_{X}\right)g:(Xdef _,OXdef )→(X_,OX) of locally ringed sites.
Let us now describe the promised algebraization results. The classical Chow's theorem states that a closed complex analytic subset Z Z ZZZ of X an X an  X^("an ")X^{\text {an }}Xan  for X X XXX smooth projective over C C C\mathbb{C}C is in fact algebraic. This fails dramatically if X X XXX is only quasiprojective, as shown by the graph of the complex exponential in ( A 2 ) an A 2 an  (A^(2))^("an ")\left(\mathbb{A}^{2}\right)^{\text {an }}(A2)an . However, Peterzil and Starchenko, generalizing [36] in the R alg R alg  R_("alg ")\mathbb{R}_{\text {alg }}Ralg  case, have shown the following:
Theorem 3.10 ([69,70]). Let X X XXX be a complex quasiprojective variety and let Z X an Z ⊂ X an  Z subX^("an ")Z \subset X^{\text {an }}Z⊂Xan  be a closed analytic subvariety. If Z Z ZZZ is definable in X def X def  X^("def ")X^{\text {def }}Xdef  then Z Z ZZZ is complex algebraic in X X XXX.
Chow's theorem, which deals only with spaces, was extended to sheaves by Serre [83]: when X X XXX is proper, the analytification functor ( ) an : Coh ( X ) Coh ( X an ) ( ⋅ ) an  : Coh ⁡ ( X ) → Coh ⁡ X an  (*)^("an "):Coh(X)rarr Coh(X^("an "))(\cdot)^{\text {an }}: \operatorname{Coh}(X) \rightarrow \operatorname{Coh}\left(X^{\text {an }}\right)(⋅)an :Coh⁡(X)→Coh⁡(Xan ) defines an equivalence of categories between the categories of coherent sheaves Coh ( X ) Coh ⁡ ( X ) Coh(X)\operatorname{Coh}(X)Coh⁡(X) and Coh ( X a n ) Coh ⁡ X a n Coh(X^(an))\operatorname{Coh}\left(X^{\mathrm{an}}\right)Coh⁡(Xan). In the definable world, let X X XXX be a separated scheme (or algebraic space) of finite type over C C C\mathbb{C}C. Associating with a coherent sheaf F F FFF on X X XXX the coherent sheaf F def := F g 1 O X O X def F def  := F ⊗ g − 1 O X O X def  F^("def "):=Fox_(g^(-1))O_(X)O_(X^("def "))F^{\text {def }}:=F \otimes_{g^{-1}} \mathcal{O}_{X} \mathcal{O}_{X^{\text {def }}}Fdef :=F⊗g−1OXOXdef  on the S S SSS-definabilization X def X def  X^("def ")X^{\text {def }}Xdef  of X X XXX, one obtains a definabilization functor (.) ) def : Coh ( X ) ) def  : Coh ⁡ ( X ) → )^("def "):Coh(X)rarr)^{\text {def }}: \operatorname{Coh}(X) \rightarrow)def :Coh⁡(X)→ Coh ( X def ) Coh ⁡ X def  Coh(X^("def "))\operatorname{Coh}\left(X^{\text {def }}\right)Coh⁡(Xdef ). Similarly there is an analytification functor X X an X ⇝ X an  X⇝X^("an ")X \rightsquigarrow X^{\text {an }}X⇝Xan  from complex definable analytic spaces to complex analytic spaces, that induces a functor ( ) a n : Coh ( X ) ( ⋅ ) a n : Coh ⁡ ( X ) → (*)^(an):Coh(X)rarr(\cdot)^{\mathrm{an}}: \operatorname{Coh}(\mathcal{X}) \rightarrow(⋅)an:Coh⁡(X)→ Coh ( X a n ) Coh ⁡ X a n Coh(X^(an))\operatorname{Coh}\left(\mathcal{X}^{\mathrm{an}}\right)Coh⁡(Xan).
Theorem 3.11 ([10]). For every separated algebraic space of finite type X X XXX, the definabilization functor ( ) d e f : Coh ( X ) Coh ( X d e f ) ( ⋅ ) d e f : Coh ⁡ ( X ) → Coh ⁡ X d e f (*)^(def):Coh(X)rarr Coh(X^(def))(\cdot)^{\mathrm{def}}: \operatorname{Coh}(X) \rightarrow \operatorname{Coh}\left(X^{\mathrm{def}}\right)(⋅)def:Coh⁡(X)→Coh⁡(Xdef) is exact and fully faithful (but it is not necessarily essentially surjective). Its essential image is stable under subobjects and subquotients.
Using Theorem 3.11 and Artin's algebraization theorem for formal modification [4], one obtains the following useful algebraization result for definable images of algebraic spaces, which will be used in Section 3.6.2:
Theorem 3.12 ([10]). Let X X XXX be a separated algebraic space of finite type and let E E E\mathcal{E}E be a definable analytic space. Any proper definable analytic map Φ : X def E Φ : X def  → E Phi:X^("def ")rarrE\Phi: X^{\text {def }} \rightarrow \mathcal{E}Φ:Xdef →E factors uniquely as f d e f ∙ ∘ f d e f ∙@f^(def)\bullet \circ f^{\mathrm{def}}∙∘fdef, where f : X Y f : X → Y f:X rarr Yf: X \rightarrow Yf:X→Y is a proper morphism of separated algebraic spaces (of finite type) such that O Y f O X O Y → f ∗ O X O_(Y)rarrf_(**)O_(X)\mathcal{O}_{Y} \rightarrow f_{*} \mathcal{O}_{X}OY→f∗OX is injective, and ι : Y def E ι : Y def  ↪ E iota:Y^("def ")↪E\iota: Y^{\text {def }} \hookrightarrow \mathcal{E}ι:Ydef ↪E is a closed immersion of definable analytic spaces.

3.4. Definability of Hodge varieties

Let us now describe the first result establishing that o-minimal geometry is potentially interesting for Hodge theory.
Theorem 3.13 ([11]). Any Hodge variety Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D can be naturally endowed with a functorial structure ( Γ D ) R alg ( Γ ∖ D ) R alg  (Gamma\\D)^(R_("alg "))(\Gamma \backslash D)^{\mathbb{R}_{\text {alg }}}(Γ∖D)Ralg  of R alg R alg  R_("alg ")\mathbb{R}_{\text {alg }}Ralg -definable complex analytic space.
Here "functorial" means that that any morphism ( G , D ) ( G , D ) G ′ , D ′ → ( G , D ) (G^('),D^('))rarr(G,D)\left(\mathbf{G}^{\prime}, D^{\prime}\right) \rightarrow(\mathbf{G}, D)(G′,D′)→(G,D) of Hodge data induces a definable map ( Γ D ) R alg ( Γ D ) R alg Γ ′ ∖ D ′ R alg  → ( Γ ∖ D ) R alg  (Gamma^(')\\D^('))^(R_("alg "))rarr(Gamma\\D)^(R_("alg "))\left(\Gamma^{\prime} \backslash D^{\prime}\right)^{\mathbb{R}_{\text {alg }}} \rightarrow(\Gamma \backslash D)^{\mathbb{R}_{\text {alg }}}(Γ′∖D′)Ralg →(Γ∖D)Ralg  of Hodge varieties. Let us sketch the construction of ( Γ D ) R alg ( Γ ∖ D ) R alg  (Gamma\\D)^(R_("alg "))(\Gamma \backslash D)^{\mathbb{R}_{\text {alg }}}(Γ∖D)Ralg . Without loss of generality (replacing G G G\mathbf{G}G by its adjoint group if necessary), we can assume that G G G\mathbf{G}G is semisimple, G = G ( R ) + G = G ( R ) + G=G(R)^(+)G=\mathbf{G}(\mathbb{R})^{+}G=G(R)+. For simplicity, let us assume that the arithmetic lattice Γ Î“ Gamma\GammaΓ is torsion free. We choose a base point in D = G / M D = G / M D=G//MD=G / MD=G/M. Notice that

making the projection G G / M G → G / M G rarr G//MG \rightarrow G / MG→G/M semialgebraic. To define an R alg R alg  R_("alg ")\mathbb{R}_{\text {alg }}Ralg -structure on Γ ( G / M ) Γ ∖ ( G / M ) Gamma\\(G//M)\Gamma \backslash(G / M)Γ∖(G/M), it is thus enough to find a semialgebraic open fundamental set F G / M F ⊂ G / M F sub G//MF \subset G / MF⊂G/M for the action of Γ Î“ Gamma\GammaΓ and to write Γ G / M = Γ F Γ ∖ G / M = Γ ∖ F Gamma\\G//M=Gamma\\F\Gamma \backslash G / M=\Gamma \backslash FΓ∖G/M=Γ∖F, where the right-hand side is the quotient of F F FFF by the closed étale semialgebraic equivalence relation induced by the action of Γ Î“ Gamma\GammaΓ on D D DDD. Here by fundamental set we mean that the set of γ Γ Î³ ∈ Γ gamma in Gamma\gamma \in \Gammaγ∈Γ such that γ F F γ F ∩ F ≠ ∅ gamma F nn F!=O/\gamma F \cap F \neq \emptysetγF∩F≠∅ is finite. We construct the fundamental set F F FFF using the reduction theory of arithmetic groups, namely the theory of Siegel sets. Let K K KKK be the unique maximal compact subgroup of G G GGG containing M M MMM. For any Q Q Q\mathbb{Q}Q-parabolic P P P\mathbf{P}P of G G G\mathbf{G}G with unipotent radical N N N\mathbf{N}N, the maximal compact subgroup K K KKK of G G GGG determines a real Levi L G L ⊂ G L sub GL \subset GL⊂G which decomposes as L = A Q L = A Q L=AQL=A QL=AQ where A A AAA is the center and Q Q QQQ is semisimple. A semialgebraic Siegel set of G G GGG associated to P P P\mathbf{P}P and K K KKK is then a set of the form S = U ( a A > 0 ) W S = U a A > 0 W S=U(aA_( > 0))W\mathbb{S}=U\left(a A_{>0}\right) WS=U(aA>0)W where U N ( R ) , W Q K U ⊂ N ( R ) , W ⊂ Q K U subN(R),W sub QKU \subset \mathbf{N}(\mathbb{R}), W \subset Q KU⊂N(R),W⊂QK are bounded semialgebraic subsets, a A a ∈ A a in Aa \in Aa∈A, and A > 0 A > 0 A_( > 0)A_{>0}A>0 is the cone corresponding to the positive root chamber. By a Siegel set of G G GGG associated to K K KKK we mean a semialgebraic Siegel set associated to P P P\mathbf{P}P and K K KKK for some Q Q Q\mathbb{Q}Q-parabolic P P P\mathbf{P}P of G G G\mathbf{G}G Suppose now that Γ G Γ ⊂ G Gamma sub G\Gamma \subset GΓ⊂G is an arithmetic group. A fundamental result of Borel [16] states that there exists finitely many Siegel sets i G , 1 i s â„‘ i ⊂ G , 1 ≤ i ≤ s â„‘_(i)sub G,1 <= i <= s\Im_{i} \subset G, 1 \leq i \leq sâ„‘i⊂G,1≤i≤s, associated with K K KKK, whose images in Γ G / K Γ ∖ G / K Gamma\\G//K\Gamma \backslash G / KΓ∖G/K cover the whole space; and such that for any 1 i j s 1 ≤ i ≠ j ≤ s 1 <= i!=j <= s1 \leq i \neq j \leq s1≤i≠j≤s, the set of γ Γ Î³ ∈ Γ gamma in Gamma\gamma \in \Gammaγ∈Γ such that γ i S j γ â„‘ i ∩ S j ≠ ∅ gammaâ„‘_(i)nnS_(j)!=O/\gamma \Im_{i} \cap \mathbb{S}_{j} \neq \emptysetγℑi∩Sj≠∅ is finite. We call the images S i , D := S i / M S i , D := S i / M S_(i,D):=S_(i)//M\mathbb{S}_{i, D}:=\mathbb{S}_{i} / MSi,D:=Si/M Siegel sets for D D DDD. Noticing that these Siegel sets for D D DDD are semialgebraic in D D DDD, we can take F = i = 1 s i , D F = ∐ i = 1 s   ⊆ i , D F=∐_(i=1)^(s)sube_(i,D)F=\coprod_{i=1}^{s} \subseteq_{i, D}F=∐i=1s⊆i,D. It is not difficult to show that the R alg R alg  R_("alg ")\mathbb{R}_{\text {alg }}Ralg -structure thus constructed is independent of the choice of the base point e M G / M e M ∈ G / M eM in G//Me M \in G / MeM∈G/M. The functoriality follows from a nontrivial property of Siegel sets with respect to morphisms of algebraic groups, due to Orr [68].

3.5. Definability of period maps

Once Theorem 3.13 is in place, the following result shows that o-minimal geometry is a natural framework for Hodge theory:
Theorem 3.14 ([11]). Let S S SSS be a smooth connected complex quasiprojective variety. Any period map Φ : S a n Γ D Φ : S a n → Γ ∖ D Phi:S^(an)rarr Gamma\\D\Phi: S^{\mathrm{an}} \rightarrow \Gamma \backslash DΦ:San→Γ∖D is the analytification of a morphism Φ : S R a n , e x p ( Γ D ) R a n , e x p Φ : S R a n , e x p → ( Γ ∖ D ) R a n , e x p Phi:S^(R_(an,exp))rarr(Gamma\\D)^(R_(an,exp))\Phi: S^{\mathbb{R}_{\mathrm{an}, \mathrm{exp}}} \rightarrow(\Gamma \backslash D)^{\mathbb{R}_{\mathrm{an}, \mathrm{exp}}}Φ:SRan,exp→(Γ∖D)Ran,exp of R a n , exp R a n , exp R_(an,exp)\mathbb{R}_{\mathrm{an}, \exp }Ran,exp-definable complex analytic spaces, where the R a n , exp R a n , exp R_(an,exp)\mathbb{R}_{\mathrm{an}, \exp }Ran,exp-structures on S ( C ) S ( C ) S(C)S(\mathbb{C})S(C) and Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D extend their natural R alg R alg  R_("alg ")\mathbb{R}_{\text {alg }}Ralg -structures defined in Example 3.7 and Theorem 3.13, respectively.
In down-to-earth terms, this means that we can cover S S SSS by finitely many open affine charts S i S i S_(i)S_{i}Si such that Φ Î¦ Phi\PhiΦ restricted to ( Res C / R S i ) ( R ) = S i ( C ) Res C / R ⁡ S i ( R ) = S i ( C ) (Res_(C//R)S_(i))(R)=S_(i)(C)\left(\operatorname{Res}_{\mathbb{C} / \mathbb{R}} S_{i}\right)(\mathbb{R})=S_{i}(\mathbb{C})(ResC/R⁡Si)(R)=Si(C) and read in a chart of Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D defined by a Siegel set of D D DDD, can be written using only real polynomials, the real exponential function, and restricted real analytic functions! This statement is already nontrivial when S = S = S=S=S= Sh is a Shimura variety and Φ a n : S a n Γ D Φ a n : S a n → Γ ∖ D Phi^(an):S^(an)rarr Gamma\\D\Phi^{\mathrm{an}}: S^{\mathrm{an}} \rightarrow \Gamma \backslash DΦan:San→Γ∖D is the identity map coming from the uniformization π : D S an Ï€ : D → S an  pi:D rarrS^("an ")\pi: D \rightarrow S^{\text {an }}Ï€:D→San  of S an S an  S^("an ")S^{\text {an }}San  by the hermitian symmetric domain D = G / K D = G / K D=G//KD=G / KD=G/K. In that case the R alg R alg  R_("alg ")\mathbb{R}_{\text {alg }}Ralg -definable varieties S h R alg S h R alg  Sh^(R_("alg "))\mathrm{Sh}^{\mathbb{R}_{\text {alg }}}ShRalg  and ( Γ D ) R alg ( Γ ∖ D ) R alg  (Gamma\\D)^(R_("alg "))(\Gamma \backslash D)^{\mathbb{R}_{\text {alg }}}(Γ∖D)Ralg  are not isomorphic, but Theorem 3.14 claims that their R a n , exp R a n , exp R_(an,exp)\mathbb{R}_{\mathrm{an}, \exp }Ran,exp-extensions Sh R a n , e x p Sh R a n , e x p Sh^(R_(an,exp))\operatorname{Sh}^{\mathbb{R}_{\mathrm{an}, \mathrm{exp}}}ShRan,exp and ( Γ D ) R a n , exp ( Γ ∖ D ) R a n ,  exp  (Gamma\\D)^(R_(an," exp "))(\Gamma \backslash D)^{\mathbb{R}_{\mathrm{an}, \text { exp }}}(Γ∖D)Ran, exp  are. This is equivalent to showing that the restriction π D : S D S R a n , exp Ï€ ∣ â„‘ D : S D → S R a n ,  exp  pi_(∣ℑ_(D)):S_(D)rarrS^(R_(an," exp "))\pi_{\mid \Im_{D}}: \mathbb{S}_{D} \rightarrow S^{\mathbb{R}_{\mathrm{an}, \text { exp }}}π∣ℑD:SD→SRan, exp  to a Siegel set for D D DDD can be written using only real polynomials, the real exponential function, and restricted real analytic functions. This
is a nice exercise on the j j jjj-function when Sh is a modular curve, was done in [71] and [76] for S h = A g S h = A g Sh=A_(g)\mathrm{Sh}=\mathscr{A}_{g}Sh=Ag, and [58] in general.
Let us sketch the proof of Theorem 3.14. We choose a log-smooth compactification of S S SSS, hence providing us with a definable cover of S R an S R an  S^(R_("an "))S^{\mathbb{R}_{\text {an }}}SRan  by punctured polydisks ( Δ ) k × Δ l Δ ∗ k × Δ l (Delta^(**))^(k)xxDelta^(l)\left(\Delta^{*}\right)^{k} \times \Delta^{l}(Δ∗)k×Δl. We are reduced to showing that the restriction of Φ Î¦ Phi\PhiΦ to such a punctured polydisk is R a n , exp R a n , exp − R_(an,exp^(-))\mathbb{R}_{\mathrm{an}, \exp ^{-}}Ran,exp− definable. This is clear if k = 0 k = 0 k=0k=0k=0, as in this case φ : Δ k + l Γ D φ : Δ k + l → Γ ∖ D varphi:Delta^(k+l)rarr Gamma\\D\varphi: \Delta^{k+l} \rightarrow \Gamma \backslash Dφ:Δk+l→Γ∖D is even R an R an  R_("an ")\mathbb{R}_{\text {an }}Ran -definable. For k > 0 k > 0 k > 0k>0k>0, let e : exp ( 2 π i ) : S Δ e : exp ⁡ ( 2 Ï€ i â‹… ) : S → Δ ∗ e:exp(2pii*):SrarrDelta^(**)\mathrm{e}: \exp (2 \pi \mathrm{i} \cdot): \mathfrak{S} \rightarrow \Delta^{*}e:exp⁡(2Ï€iâ‹…):S→Δ∗ be the universal covering map. Its restriction to a sufficiently large bounded vertical strip V := [ a , b ] × ] 0 , + [ N = { x + i y , y > 0 } V := [ a , b ] × ] 0 , + ∞ [ ⊂ N = { x + i y , y > 0 } V:=[a,b]xx]0,+oo[subN={x+iy,y > 0}V:=[a, b] \times] 0,+\infty[\subset \mathfrak{N}=\{x+\mathrm{i} y, y>0\}V:=[a,b]×]0,+∞[⊂N={x+iy,y>0} is R a n , e x p R a n , e x p R_(an,exp)\mathbb{R}_{\mathrm{an}, \mathrm{exp}}Ran,exp-definable. Considering the following commutative diagram:
it is thus enough to show that π Φ ~ : V k × Δ l Γ D Ï€ ∘ Φ ~ : V k × Δ l → Γ ∖ D pi@ widetilde(Phi):V^(k)xxDelta^(l)rarr Gamma\\D\pi \circ \widetilde{\Phi}: V^{k} \times \Delta^{l} \rightarrow \Gamma \backslash Dπ∘Φ~:Vk×Δl→Γ∖D is R an,exp R an,exp  R_("an,exp ")\mathbb{R}_{\text {an,exp }}Ran,exp -definable.
Let the coordinates of ( Δ ) k × Δ l Δ ∗ k × Δ l (Delta^(**))^(k)xxDelta^(l)\left(\Delta^{*}\right)^{k} \times \Delta^{l}(Δ∗)k×Δl be t i , 1 i k + l t i , 1 ≤ i ≤ k + l t_(i),1 <= i <= k+lt_{i}, 1 \leq i \leq k+lti,1≤i≤k+l, those of S k S k S^(k)\mathscr{S}^{k}Sk be z i , 1 i k z i , 1 ≤ i ≤ k z_(i),1 <= i <= kz_{i}, 1 \leq i \leq kzi,1≤i≤k, so that e ( z i ) = t i e z i = t i e(z_(i))=t_(i)\mathrm{e}\left(z_{i}\right)=t_{i}e(zi)=ti. Let T i T i T_(i)T_{i}Ti be the monodromy at infinity of Φ Î¦ Phi\PhiΦ around the hyperplane ( z i = 0 ) z i = 0 (z_(i)=0)\left(z_{i}=0\right)(zi=0), boundary component of S ¯ S S ¯ ∖ S bar(S)\\S\bar{S} \backslash SS¯∖S. By Borel's theorem T i T i T_(i)T_{i}Ti is quasiunipotent. Replacing S S SSS by a finite étale cover, we can without loss of generality assume that each T i = exp ( N i ) T i = exp ⁡ N i T_(i)=exp(N_(i))T_{i}=\exp \left(N_{i}\right)Ti=exp⁡(Ni), with N i N i ∈ N_(i)inN_{i} \inNi∈ g g g\mathrm{g}g nilpotent. The Nilpotent Orbit Theorem of Schmid is equivalent to saying that Φ ~ : V k × Î¦ ~ : V k × widetilde(Phi):V^(k)xx\widetilde{\Phi}: V^{k} \timesΦ~:Vk× Δ l D Δ l → D Delta^(l)rarr D\Delta^{l} \rightarrow DΔl→D can be written as Φ ~ ( z 1 , , z k , t k + 1 , , t k + l ) = exp ( i = 1 k z i N i ) Ψ ( t 1 , , t k + l ) Φ ~ z 1 , … , z k , t k + 1 , … , t k + l = exp ⁡ ∑ i = 1 k   z i N i â‹… Ψ t 1 , … , t k + l widetilde(Phi)(z_(1),dots,z_(k),t_(k+1),dots,t_(k+l))=exp(sum_(i=1)^(k)z_(i)N_(i))*Psi(t_(1),dots,t_(k+l))\widetilde{\Phi}\left(z_{1}, \ldots, z_{k}, t_{k+1}, \ldots, t_{k+l}\right)=\exp \left(\sum_{i=1}^{k} z_{i} N_{i}\right) \cdot \Psi\left(t_{1}, \ldots, t_{k+l}\right)Φ~(z1,…,zk,tk+1,…,tk+l)=exp⁡(∑i=1kziNi)⋅Ψ(t1,…,tk+l) for Ψ : Δ k × Δ l D ˇ an Ψ : Δ k × Δ l → D ˇ an  Psi:Delta^(k)xxDelta^(l)rarrD^(ˇ)^("an ")\Psi: \Delta^{k} \times \Delta^{l} \rightarrow \check{D}^{\text {an }}Ψ:Δk×Δl→Dˇan  a holomorphic map. On the one hand, Ψ Î¨ Psi\PsiΨ is R an R an  R_("an ")\mathbb{R}_{\text {an }}Ran -definable as a

i k i ≤ k i <= ki \leq ki≤k, and the variables t j , k + 1 j k + l t j , k + 1 ≤ j ≤ k + l t_(j),k+1 <= j <= k+lt_{j}, k+1 \leq j \leq k+ltj,k+1≤j≤k+l. On the other hand, exp ( i = 1 k z i N i ) exp ⁡ ∑ i = 1 k   z i N i ∈ exp(sum_(i=1)^(k)z_(i)N_(i))in\exp \left(\sum_{i=1}^{k} z_{i} N_{i}\right) \inexp⁡(∑i=1kziNi)∈ G ( C ) G ( C ) G(C)\mathbf{G}(\mathbb{C})G(C) is polynomial in the variables z i z i z_(i)z_{i}zi, as the monodromies N i N i N_(i)N_{i}Ni are nilpotent and commute pairwise. As the action of G ( C ) G ( C ) G(C)\mathbf{G}(\mathbb{C})G(C) on D ˇ D ˇ D^(ˇ)\check{D}Dˇ is algebraic, it follows that Φ ~ : V k × Δ l D Φ ~ : V k × Δ l → D widetilde(Phi):V^(k)xxDelta^(l)rarr D\widetilde{\Phi}: V^{k} \times \Delta^{l} \rightarrow DΦ~:Vk×Δl→D is R a n , exp R a n , exp R_(an,exp)\mathbb{R}_{\mathrm{an}, \exp }Ran,exp-definable. The proof of Theorem 3.14 is thus reduced to the following, proven by Schmid when k = 1 , l = 0 [ 82 , 5.29 ] k = 1 , l = 0 [ 82 , 5.29 ] k=1,l=0[82,5.29]k=1, l=0[82,5.29]k=1,l=0[82,5.29] :
Theorem 3.15 ([11]). The image Φ ~ ( V k × Δ l ) Φ ~ V k × Δ l widetilde(Phi)(V^(k)xxDelta^(l))\widetilde{\Phi}\left(V^{k} \times \Delta^{l}\right)Φ~(Vk×Δl) lies in a finite union of Siegel sets of D D DDD.
This can be interpreted as showing that, possibly after passing to a definable cover of V k × Δ l V k × Δ l V^(k)xxDelta^(l)V^{k} \times \Delta^{l}Vk×Δl, the Hodge form of Φ ~ Φ ~ widetilde(Phi)\widetilde{\Phi}Φ~ is Minkowski reduced with respect to a flat frame. This is done using the hard analytic theory of Hodge forms estimates for degenerations of variations of Hodge structure, as in [53, THEOREMS 3.4.1 AND 3.4.2] and [23, THEOREM 5.21].
Remark 3.16. Theorems 3.13 and 3.14 have been extended to the mixed case in [9].

3.6. Applications

3.6.1. About the Cattani-Deligne-Kaplan theorem

As a corollary of Theorems 3.14 and 3.10 one obtains the following, which, in view of (2.3), implies immediately Theorem 3.1 :
Theorem 3.17 ([11]). Let S S SSS be a smooth quasiprojective complex variety. Let V V V\mathbb{V}V be a polarized Z V H S Z V H S ZVHS\mathbb{Z} V H SZVHS on S a n S a n S^(an)S^{\mathrm{an}}San with period map Φ : S a n Γ D Φ : S a n → Γ ∖ D Phi:S^(an)rarr Gamma\\D\Phi: S^{\mathrm{an}} \rightarrow \Gamma \backslash DΦ:San→Γ∖D. For any special subvariety Γ D Γ ′ ∖ D ′ ⊂ Gamma^(')\\D^(')sub\Gamma^{\prime} \backslash D^{\prime} \subsetΓ′∖D′⊂ Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D, its preimage Φ 1 ( Γ D ) Φ − 1 Γ ′ ∖ D ′ Phi^(-1)(Gamma^(')\\D^('))\Phi^{-1}\left(\Gamma^{\prime} \backslash D^{\prime}\right)Φ−1(Γ′∖D′) is a finite union of irreducible algebraic subvarieties of S S SSS.
Indeed, it follows from Theorem 3.13 that Γ D Γ ′ ∖ D ′ Gamma^(')\\D^(')\Gamma^{\prime} \backslash D^{\prime}Γ′∖D′ is definable in ( Γ D ) R alg ( Γ ∖ D ) R alg  (Gamma\\D)^(R_("alg "))(\Gamma \backslash D)^{\mathbb{R}_{\text {alg }}}(Γ∖D)Ralg . By Theorem 3.14, its preimage Φ 1 ( Γ D ) Φ − 1 Γ ′ ∖ D ′ Phi^(-1)(Gamma^(')\\D^('))\Phi^{-1}\left(\Gamma^{\prime} \backslash D^{\prime}\right)Φ−1(Γ′∖D′) is definable in S R annexp S R annexp  S^(R_("annexp "))S^{\mathbb{R}_{\text {annexp }}}SRannexp . As Φ Î¦ Phi\PhiΦ is holomorphic and Γ D Γ D Γ ′ ∖ D ′ ⊂ Γ ∖ D Gamma^(')\\D^(')sub Gamma\\D\Gamma^{\prime} \backslash D^{\prime} \subset \Gamma \backslash DΓ′∖D′⊂Γ∖D is a closed complex analytic subvariety, Φ 1 ( Γ D ) Φ − 1 Γ ′ ∖ D ′ Phi^(-1)(Gamma^(')\\D^('))\Phi^{-1}\left(\Gamma^{\prime} \backslash D^{\prime}\right)Φ−1(Γ′∖D′) is also a closed complex analytic subvariety of S an S an  S^("an ")S^{\text {an }}San . By Theorem 3.10, it is thus algebraic in S S SSS.
Remark 3.18. Theorem 3.17 has been extended to the mixed case in [9], thus recovering [ 18 21 ] [ 18 − 21 ] [18-21][18-21][18−21].
Let Y S Y ⊂ S Y sub SY \subset SY⊂S be a closed irreducible algebraic subvariety. Let ( G Y , D Y ) ( G , D ) G Y , D Y ⊂ ( G , D ) (G_(Y),D_(Y))sub(G,D)\left(\mathbf{G}_{Y}, D_{Y}\right) \subset(\mathbf{G}, D)(GY,DY)⊂(G,D) be the generic Hodge datum of V V V\mathbb{V}V restricted to the smooth locus of of Y Y YYY. There exist a smallest Hodge subvariety Γ Y D Y Γ Y ∖ D Y Gamma_(Y)\\D_(Y)\Gamma_{Y} \backslash D_{Y}ΓY∖DY of Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D containing Φ ( Y an ) Φ Y an  Phi(Y^("an "))\Phi\left(Y^{\text {an }}\right)Φ(Yan ). The following terminology will be convenient:
Definition 3.19. Let S S SSS be a smooth quasiprojective complex variety. Let V V V\mathbb{V}V be a polarized Z Z Z\mathbb{Z}Z VHS on S an S an  S^("an ")S^{\text {an }}San  with period map Φ : S an Γ D Φ : S an  → Γ ∖ D Phi:S^("an ")rarr Gamma\\D\Phi: S^{\text {an }} \rightarrow \Gamma \backslash DΦ:San →Γ∖D. A closed irreducible subvariety Y S Y ⊂ S Y sub SY \subset SY⊂S is called a special subvariety of S S SSS for V V V\mathbb{V}V if it coincides with an irreducible component of the preimage Φ 1 ( Γ Y D Y ) Φ − 1 Γ Y ∖ D Y Phi^(-1)(Gamma_(Y)\\D_(Y))\Phi^{-1}\left(\Gamma_{Y} \backslash D_{Y}\right)Φ−1(ΓY∖DY).
Equivalently, a special subvariety of S S SSS for V V V\mathbb{V}V is a closed irreducible algebraic subvariety Y S Y ⊂ S Y sub SY \subset SY⊂S maximal among the closed irreducible algebraic subvarieties Z Z ZZZ of S S SSS such that the generic Mumford-Tate group G Z G Z G_(Z)\mathbf{G}_{Z}GZ of V Z V ∣ Z V_(∣Z)\mathbb{V}_{\mid Z}V∣Z equals G Y G Y G_(Y)\mathbf{G}_{Y}GY.

3.6.2. A conjecture of Griffiths

Combining Theorem 3.14 this time with Theorem 3.12 leads to a proof of an old conjecture of Griffiths [44], claiming that the image of any period map has a natural structure of quasiprojective variety (Griffiths proved it when the target Hodge variety is compact):
Theorem 3.20 ([10]). Let S S SSS be a smooth connected quasiprojective complex variety and let Φ : S a n Γ D Φ : S a n → Γ ∖ D Phi:S^(an)rarr Gamma\\D\Phi: S^{\mathrm{an}} \rightarrow \Gamma \backslash DΦ:San→Γ∖D be a period map. There exists a unique dominant morphism of complex algebraic varieties f : S T f : S → T f:S rarr Tf: S \rightarrow Tf:S→T, with T T TTT quasiprojective, and a closed complex analytic immersion ι : T a n Γ D ι : T a n ↪ Γ ∖ D iota:T^(an)↪Gamma\\D\iota: T^{\mathrm{an}} \hookrightarrow \Gamma \backslash Dι:Tan↪Γ∖D such that Φ = ι f a n Φ = ι ∘ f a n Phi=iota@f^(an)\Phi=\iota \circ f^{\mathrm{an}}Φ=ι∘fan.
Let us sketch the proof. As before, let S S ¯ S ⊂ S ¯ S sub bar(S)S \subset \bar{S}S⊂S¯ be a log-smooth compactification by a simple normal crossing divisor Z Z ZZZ. It follows from a result of Griffiths [43, PROP. 9.11I)] that Φ Î¦ Phi\PhiΦ extends to a proper period map over the components of Z Z ZZZ around which the monodromy is finite. Hence, without loss of generality, we can assume that Φ Î¦ Phi\PhiΦ is proper. The existence of f f fff in the category of algebraic spaces then follows immediately from Theorems 3.14 and 3.12 (for S = R a n ,exp S = R a n ,exp  S=R_(an",exp ")S=\mathbb{R}_{\mathrm{an} \text {,exp }}S=Ran,exp  ). The proof that T T TTT is in fact quasiprojective exploits a crucial observation of Griffiths that Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D carries a positively curved Q Q Q\mathbb{Q}Q-line bundle L := p det ( F p ) L := ⊗ p det ⁡ F p L:=ox_(p)det(F^(p))\mathscr{L}:=\otimes_{p} \operatorname{det}\left(F^{p}\right)L:=⊗pdet⁡(Fp). This line bundle is naturally definable on ( Γ D ) def ( Γ ∖ D ) def  (Gamma\\D)^("def ")(\Gamma \backslash D)^{\text {def }}(Γ∖D)def . Using the definable GAGA Theorem 3.11, one shows that its restriction to T def T def  T^("def ")T^{\text {def }}Tdef  comes from an algebraic Q Q Q\mathbb{Q}Q-line bundle L T L T L_(T)L_{T}LT on T T TTT, which one manages to show to be ample.

4. FUNCTIONAL TRANSCENDENCE

4.1. Bialgebraic geometry

As we saw, Hodge theory, which compares the Hodge filtration on H d R ( X / C ) H d R ∙ ( X / C ) H_(dR)^(∙)(X//C)H_{\mathrm{dR}}^{\bullet}(X / \mathbb{C})HdR∙(X/C) with the rational structure on H B ( X a n , C ) H B ∙ X a n , C H_(B)^(∙)(X^(an),C)H_{\mathrm{B}}^{\bullet}\left(X^{\mathrm{an}}, \mathbb{C}\right)HB∙(Xan,C), gives rise to variational Hodge theory, whose fundamental diagram (2.2) compares the algebraic structure of S S SSS with the algebraic structure on the dual period domain D ˇ D ˇ D^(ˇ)\check{D}Dˇ. As such, it is a partial answer to one of the most classical problem of complex algebraic geometry: the transcendental nature of the topological universal cover of complex algebraic varieties. If S S SSS is a connected complex algebraic variety, the universal cover S a n ~ S a n ~ widetilde(S^(an))\widetilde{S^{\mathrm{an}}}San~ has usually no algebraic structure as soon as the topological fundamental group π 1 ( S a n ) Ï€ 1 S a n pi_(1)(S^(an))\pi_{1}\left(S^{\mathrm{an}}\right)Ï€1(San) is infinite. As an aside, let us mention an interesting conjecture of Kóllar and Pardon [60], predicting that if X X XXX is a normal projective irreducible complex variety whose universal cover X an ~ X an  ~ widetilde(X^("an "))\widetilde{X^{\text {an }}}Xan ~ is biholomorphic to a semialgebraic open subset of an algebraic variety then X an ~ X an  ~ widetilde(X^("an "))\widetilde{X^{\text {an }}}Xan ~ is biholomorphic to C n × D × F an C n × D × F an  C^(n)xx D xxF^("an ")\mathbb{C}^{n} \times D \times F^{\text {an }}Cn×D×Fan , where D D DDD is a bounded symmetric domain and F F FFF is a normal, projective, irreducible, topologically simply connected, complex algebraic variety We want to think of variational Hodge theory as an attempt to provide a partial algebraic uniformization: the period map emulates an algebraic structure on S a n ~ S a n ~ widetilde(S^(an))\widetilde{S^{\mathrm{an}}}San~, modeled on the flag variety D ˇ D ˇ D^(ˇ)\check{D}Dˇ. The remaining task is then to describe the transcendence properties of the complex analytic uniformization map p : S a n ~ S a n p : S a n ~ → S a n p: widetilde(S^(an))rarrS^(an)p: \widetilde{S^{\mathrm{an}}} \rightarrow S^{\mathrm{an}}p:San~→San with respects to the emulated algebraic structure on S a n ~ S a n ~ widetilde(S^(an))\widetilde{S^{\mathrm{an}}}San~ and the algebraic structure S S SSS on S an S an  S^("an ")S^{\text {an }}San . A few years ago, the author [55], together with Ullmo and Yafaev [59], introduced a convenient format for studying such questions, which encompasses many classical transcendence problems and provides a powerful heuristic.
Definition 4.1. A bialgebraic structure on a connected quasiprojective variety S S SSS over C C C\mathbb{C}C is a pair
( f : S a n ~ Z a n , ρ : π 1 ( S a n ) Aut ( Z ) ) f : S a n ~ → Z a n , ρ : Ï€ 1 S a n → Aut ⁡ ( Z ) (f:( widetilde(S^(an)))rarrZ^(an),rho:pi_(1)(S^(an))rarr Aut(Z))\left(f: \widetilde{S^{\mathrm{an}}} \rightarrow Z^{\mathrm{an}}, \rho: \pi_{1}\left(S^{\mathrm{an}}\right) \rightarrow \operatorname{Aut}(Z)\right)(f:San~→Zan,ρ:Ï€1(San)→Aut⁡(Z))
where Z Z ZZZ denotes an algebraic variety (called the algebraic model of S a n ~ S a n ~ widetilde(S^(an))\widetilde{S^{\mathrm{an}}}San~ ), Aut ( Z ) Aut ⁡ ( Z ) Aut(Z)\operatorname{Aut}(Z)Aut⁡(Z) is its group of algebraic automorphisms, ρ ρ rho\rhoρ is a group morphism (called the monodromy representation) and f f fff is a ρ ρ rho\rhoρ-equivariant holomorphic map (called the developing map).
An irreducible analytic subvariety Y S a n ~ Y ⊂ S a n ~ Y sub widetilde(S^(an))Y \subset \widetilde{S^{\mathrm{an}}}Y⊂San~ is said to be an algebraic subvariety of S an ~ S an  ~ widetilde(S^("an "))\widetilde{S^{\text {an }}}San ~ for the bialgebraic structure ( f , ρ ) ( f , ρ ) (f,rho)(f, \rho)(f,ρ) if Y Y YYY is an analytic irreducible component of f 1 ( f ( Y ) ¯ Zar f − 1 f ( Y ) ¯ Zar  f^(-1)( bar(f(Y))^("Zar "):}f^{-1}\left(\overline{f(Y)}^{\text {Zar }}\right.f−1(f(Y)¯Zar  ) (where f ( Y ) ¯ Zar f ( Y ) ¯ Zar  bar(f(Y))^("Zar ")\overline{f(Y)}^{\text {Zar }}f(Y)¯Zar  denotes the Zariski-closure of f ( Y ) f ( Y ) f(Y)f(Y)f(Y) in Z Z ZZZ ). An irreducible algebraic subvariety Y S a n ~ Y ⊂ S a n ~ Y sub widetilde(S^(an))Y \subset \widetilde{S^{\mathrm{an}}}Y⊂San~, resp. W S W ⊂ S W sub SW \subset SW⊂S, is said to be bialgebraic if p ( Y ) p ( Y ) p(Y)p(Y)p(Y) is an algebraic subvariety of S S SSS, resp. any (equivalently one) analytic irreducible component of p 1 ( W ) p − 1 ( W ) p^(-1)(W)p^{-1}(W)p−1(W) is an irreducible algebraic subvariety of S a n ~ S a n ~ widetilde(S^(an))\widetilde{S^{\mathrm{an}}}San~. The bialgebraic subvarieties of S S SSS are precisely the ones where the emulated algebraic structure on S a n ~ S a n ~ widetilde(S^(an))\widetilde{S^{\mathrm{an}}}San~ and the one on S S SSS interact nontrivially.
Example 4.2. (a) tori, S = ( C ) n S = C ∗ n S=(C^(**))^(n)S=\left(\mathbb{C}^{*}\right)^{n}S=(C∗)n. The uniformization map is the multiexponential
p := ( exp ( 2 π i ) , , exp ( 2 π i ) ) : C n ( C ) n p := ( exp ⁡ ( 2 Ï€ i â‹… ) , … , exp ⁡ ( 2 Ï€ i â‹… ) ) : C n → C ∗ n p:=(exp(2pi i*),dots,exp(2pi i*)):C^(n)rarr(C^(**))^(n)p:=(\exp (2 \pi i \cdot), \ldots, \exp (2 \pi i \cdot)): \mathbb{C}^{n} \rightarrow\left(\mathbb{C}^{*}\right)^{n}p:=(exp⁡(2Ï€iâ‹…),…,exp⁡(2Ï€iâ‹…)):Cn→(C∗)n
and f f fff is the identity morphism of C n C n C^(n)\mathbb{C}^{n}Cn. An irreducible algebraic subvariety Y C n Y ⊂ C n Y subC^(n)Y \subset \mathbb{C}^{n}Y⊂Cn (resp. W ( C ) n ) W ⊂ C ∗ n {:W sub(C^(**))^(n))\left.W \subset\left(\mathbb{C}^{*}\right)^{n}\right)W⊂(C∗)n) is bialgebraic if and only if Y Y YYY is a translate of a rational linear subspace of C n = Q n Q C C n = Q n ⊗ Q C C^(n)=Q^(n)ox_(Q)C\mathbb{C}^{n}=\mathbb{Q}^{n} \otimes_{\mathbb{Q}} \mathbb{C}Cn=Qn⊗QC (resp. W W WWW is a translate of a subtorus of ( C ) n ) C ∗ n {:(C^(**))^(n))\left.\left(\mathbb{C}^{*}\right)^{n}\right)(C∗)n)
(b) abelian varieties, S = A S = A S=AS=AS=A is a complex abelian variety of dimension n n nnn. Let p : p : p:p:p: Lie A A ≃ A≃A \simeqA≃ C n A C n → A C^(n)rarr A\mathbb{C}^{n} \rightarrow ACn→A be the uniformizing map of a complex abelian variety A A AAA of dimension n n nnn. Once more S a n ~ = C n S a n ~ = C n widetilde(S^(an))=C^(n)\widetilde{S^{\mathrm{an}}}=\mathbb{C}^{n}San~=Cn and f f fff is the identity morphism. One checks easily that an irreducible algebraic subvariety W A W ⊂ A W sub AW \subset AW⊂A is bialgebraic if and only if W W WWW is the translate of an abelian subvariety of A A AAA.
(c) Shimura varieties, ( G , D ) ( G , D ) (G,D)(\mathbf{G}, D)(G,D) is a Shimura datum. The quotient S an = Γ D S an  = Γ ∖ D S^("an ")=Gamma\\DS^{\text {an }}=\Gamma \backslash DSan =Γ∖D (for Γ G := Γ ⊂ G := Gamma sub G:=\Gamma \subset G:=Γ⊂G:= G der ( R ) + G der  ( R ) + G^("der ")(R)^(+)\mathbf{G}^{\text {der }}(\mathbb{R})^{+}Gder (R)+a congruence torsion-free lattice) is the complex analytification of a (connected) Shimura variety Sh, defined over a number field (a finite extension of the reflex field of ( G , D ) ( G , D ) (G,D)(\mathbf{G}, D)(G,D) ). And f f fff is the open embedding D D an D ↪ D an  D↪D^("an ")D \hookrightarrow D^{\text {an }}D↪Dan .
Let us come back to the case of the bialgebraic structure on S S SSS
( Φ ~ : S a n ~ D ˇ a n , ρ : π 1 ( S a n ) Γ G ( Q ) ) Φ ~ : S a n ~ → D ˇ a n , ρ : Ï€ 1 S a n → Γ ⊂ G ( Q ) (( widetilde(Phi)):( widetilde(S^(an)))rarrD^(ˇ)^(an),rho:pi_(1)(S^(an))rarr Gamma subG(Q))\left(\widetilde{\Phi}: \widetilde{S^{\mathrm{an}}} \rightarrow \check{D}^{\mathrm{an}}, \rho: \pi_{1}\left(S^{\mathrm{an}}\right) \rightarrow \Gamma \subset \mathbf{G}(\mathbb{Q})\right)(Φ~:San~→Dˇan,ρ:Ï€1(San)→Γ⊂G(Q))
defined by a polarized Z V H S V Z V H S V ZVHSV\mathbb{Z V H S} \mathbb{V}ZVHSV and its period map Φ : S an Γ D Φ : S an  → Γ ∖ D Phi:S^("an ")rarr Gamma\\D\Phi: S^{\text {an }} \rightarrow \Gamma \backslash DΦ:San →Γ∖D with monodromy ρ ρ rho\rhoρ : π 1 ( S a n ) Γ G ( Q ) Ï€ 1 S a n → Γ ⊂ G ( Q ) pi_(1)(S^(an))rarr Gamma subG(Q)\pi_{1}\left(S^{\mathrm{an}}\right) \rightarrow \Gamma \subset \mathbf{G}(\mathbb{Q})Ï€1(San)→Γ⊂G(Q) (in fact, all the examples above are of this form if we consider more generally graded-polarized variations of mixed Z Z Z\mathbb{Z}Z-Hodge structures). What are its bialgebraic subvarieties? To answer this question, we need to define the weakly special subvarieties of Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D, as either a special subvariety or a subvariety of the form
Γ H D H × { t } Γ H D H × Γ L D L Γ D Γ H ∖ D H × { t } ⊂ Γ H ∖ D H × Γ L ∖ D L ⊂ Γ ∖ D Gamma_(H)\\D_(H)xx{t}subGamma_(H)\\D_(H)xxGamma_(L)\\D_(L)sub Gamma\\D\Gamma_{\mathbf{H}} \backslash D_{H} \times\{t\} \subset \Gamma_{\mathbf{H}} \backslash D_{H} \times \Gamma_{\mathbf{L}} \backslash D_{L} \subset \Gamma \backslash DΓH∖DH×{t}⊂ΓH∖DH×ΓL∖DL⊂Γ∖D
where ( H × L , D H × D L ) H × L , D H × D L (HxxL,D_(H)xxD_(L))\left(\mathbf{H} \times \mathbf{L}, D_{H} \times D_{L}\right)(H×L,DH×DL) is a Hodge subdatum of ( G ad , D ) G ad  , D (G^("ad "),D)\left(\mathbf{G}^{\text {ad }}, D\right)(Gad ,D) and { t } { t } {t}\{t\}{t} is a Hodge generic point in Γ L D L Γ L ∖ D L Gamma_(L)\\D_(L)\Gamma_{\mathbf{L}} \backslash D_{L}ΓL∖DL. Generalizing Theorem 3.17, the preimage under Φ Î¦ Phi\PhiΦ of any weakly special subvariety of Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D is an algebraic subvariety of S S SSS [56]. An irreducible component of such a preimage is called a weakly special subvariety of S S SSS for V V V\mathbb{V}V (or Φ Î¦ Phi\PhiΦ ).
Theorem 4.3 ([56]). Let Φ : S a n Γ D Φ : S a n → Γ ∖ D Phi:S^(an)rarr Gamma\\D\Phi: S^{\mathrm{an}} \rightarrow \Gamma \backslash DΦ:San→Γ∖D be a period map. The bialgebraic subvarieties of S S SSS for the bialgebraic structure defined by Φ Î¦ Phi\PhiΦ are precisely the weakly special subvarieties of S S SSS for Φ Î¦ Phi\PhiΦ. In analogy with Definition 3.19, they are also the closed irreducible algebraic subvarieties Y S Y ⊂ S Y sub SY \subset SY⊂S maximal among the closed irreducible algebraic subvarieties Z Z ZZZ of S S SSS whose algebraic monodromy group H Z H Z H_(Z)\mathbf{H}_{Z}HZ equals H Y H Y H_(Y)\mathbf{H}_{Y}HY.
When S = S h S = S h S=ShS=\mathrm{Sh}S=Sh is a Shimura variety, these results are due to Moonen [65] and [91]. In that case the weakly special subvarieties are also the irreducible algebraic subvarieties of Sh whose smooth locus is totally geodesic in S h an S h an  Sh^("an ")\mathrm{Sh}^{\text {an }}Shan  for the canonical Kähler-Einstein metric on S h an = Γ D S h an  = Γ ∖ D Sh^("an ")=Gamma\\D\mathrm{Sh}^{\text {an }}=\Gamma \backslash DShan =Γ∖D coming from the Bergman metric on D D DDD, see [65].
To study not only functional transcendence but also arithmetic transcendence, we enrich bialgebraic structures over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯. A Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯-bialgebraic structure on a quasi-projective variety S S SSS defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯ is a bialgebraic structure ( f : S a n ~ Z a n , h : π 1 ( S an ) Aut ( Z ) ) f : S a n ~ → Z a n , h : Ï€ 1 S an  → Aut ⁡ ( Z ) (f:( widetilde(S^(an)))rarrZ^(an),h:pi_(1)(S^("an "))rarr Aut(Z))\left(f: \widetilde{S^{\mathrm{an}}} \rightarrow Z^{\mathrm{an}}, h: \pi_{1}\left(S^{\text {an }}\right) \rightarrow \operatorname{Aut}(Z)\right)(f:San~→Zan,h:Ï€1(San )→Aut⁡(Z)) such that Z Z ZZZ is defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯ and the homomorphism h h hhh takes values in Aut Q ¯ Z Q ¯ Z bar(Q)Z\overline{\mathbb{Q}} ZQ¯Z. An algebraic subvariety Y S an ~ Y ⊂ S an  ~ Y sub widetilde(S^("an "))Y \subset \widetilde{S^{\text {an }}}Y⊂San ~ is said to be defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯ if its model f ( Y ) ¯ Zar Z f ( Y ) ¯ Zar  ⊂ Z bar(f(Y))^("Zar ")sub Z\overline{f(Y)}^{\text {Zar }} \subset Zf(Y)¯Zar ⊂Z is. A Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯-bialgebraic subvariety W S W ⊂ S W sub SW \subset SW⊂S is an algebraic subvariety of S S SSS defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯ and such that any (equivalently one) of the analytic irreducible components of p 1 ( W ) p − 1 ( W ) p^(-1)(W)p^{-1}(W)p−1(W) is an algebraic subvariety of
S a n ~ S a n ~ widetilde(S^(an))\widetilde{S^{\mathrm{an}}}San~ defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯. A Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯-bialgebraic point s S ( Q ¯ ) s ∈ S ( Q ¯ ) s in S( bar(Q))s \in S(\overline{\mathbb{Q}})s∈S(Q¯) is also called an arithmetic point. Example 4.2 a ) 4.2 a ) 4.2a)4.2 \mathrm{a})4.2a) is naturally defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯, with arithmetic points the torsion points of ( C ) n C ∗ n (C^(**))^(n)\left(\mathbb{C}^{*}\right)^{n}(C∗)n. In Example 4.2 b) the bialgebraic structure can be defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯ if the abelian variety A A AAA has C M C M CM\mathrm{CM}CM, and its arithmetic points are its torsion points, see [90]. Example 4.2c) is naturally a Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯-bialgebraic structure, with arithmetic points the special points of the Shimura variety (namely the special subvarieties of dimension zero), at least when the pure part of the Shimura variety is of Abelian type, see [84]. In all these cases it is interesting to notice that the Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯-bialgebraic subvarieties are the bialgebraic subvarieties containing one arithmetic point (in Example 4.2c) these are the special subvarieties of the Shimura variety).
The bi-algebraic structure associated with a period map Φ : S an Γ D Φ : S an  → Γ ∖ D Phi:S^("an ")rarr Gamma\\D\Phi: S^{\text {an }} \rightarrow \Gamma \backslash DΦ:San →Γ∖D is defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯ as soon as S S SSS is. In this case, we expect the Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯-bi-algebraic subvarieties to be precisely the special subvarieties, see [55, 2.6 AND 3.4].

4.2. The Ax-Schanuel theorem for period maps

The geometry of bialgebraic structures is controlled by the following functional transcendence heuristic, whose idea was introduced by Pila in the case of Shimura varieties, see [ 73 , 74 ] : [ 73 , 74 ] : [73,74]:[73,74]:[73,74]:
Ax-Schanuel principle. Let S S SSS be an irreducible algebraic variety endowed with a non-trivial bialgebraic structure. Let U S an ~ × S an U ⊂ S an  ~ × S an  U sub widetilde(S^("an "))xxS^("an ")U \subset \widetilde{S^{\text {an }}} \times S^{\text {an }}U⊂San ~×San  be an algebraic subvariety (for the product bialgebraic structure) and let W W WWW be an analytic irreducible component of U Δ U ∩ Δ U nn DeltaU \cap \DeltaU∩Δ, where Δ Î” Delta\DeltaΔ denotes the graph of p : S a n ~ S a n p : S a n ~ → S a n p: widetilde(S^(an))rarrS^(an)p: \widetilde{S^{\mathrm{an}}} \rightarrow S^{\mathrm{an}}p:San~→San. Then codim U W dim W ¯ b i codim U ⁡ W ≥ dim ⁡ W ¯ b i codim_(U)W >= dim bar(W)^(bi)\operatorname{codim}_{U} W \geq \operatorname{dim} \bar{W}^{\mathrm{bi}}codimU⁡W≥dim⁡W¯bi, where W ¯ b i W ¯ b i bar(W)^(bi)\bar{W}^{\mathrm{bi}}W¯bi denotes the smallest bialgebraic subvariety of S S SSS containing p ( W ) p ( W ) p(W)p(W)p(W).
When applied to a subvariety U S a n ~ × S an U ⊂ S a n ~ × S an  U sub widetilde(S^(an))xxS^("an ")U \subset \widetilde{S^{\mathrm{an}}} \times S^{\text {an }}U⊂San~×San  of the form Y × p ( Y ) ¯ Zar Y × p ( Y ) ¯ Zar  Y xx bar(p(Y))^("Zar ")Y \times \overline{p(Y)}^{\text {Zar }}Y×p(Y)¯Zar  for Y S a n ~ Y ⊂ S a n ~ Y sub widetilde(S^(an))Y \subset \widetilde{S^{\mathrm{an}}}Y⊂San~ algebraic, the Ax-Schanuel principle specializes to the following:
Ax-Lindemann principle. Let S S SSS be an irreducible algebraic variety endowed with a nontrivial bialgebraic structure. Let Y S a n ~ Y ⊂ S a n ~ Y sub widetilde(S^(an))Y \subset \widetilde{S^{\mathrm{an}}}Y⊂San~ be an algebraic subvariety. Then p ( Y ) ¯ Z a r p ( Y ) ¯ Z a r bar(p(Y))^(Zar)\overline{p(Y)}^{\mathrm{Zar}}p(Y)¯Zar is a bialgebraic subvariety of S S SSS.
Ax [ 5 , 6 ] [ 5 , 6 ] [5,6][5,6][5,6] showed that the abstract Ax-Schanuel principle holds true for Example 4.2a) and Example 4.2b) above, using differential algebra. Notice that the Ax-Lindemann principle in Example 4.2a) is the functional analog of the classical Lindemann theorem stating that if α 1 , , α n α 1 , … , α n alpha_(1),dots,alpha_(n)\alpha_{1}, \ldots, \alpha_{n}α1,…,αn are Q Q Q\mathbb{Q}Q-linearly independent algebraic numbers then e α 1 , , e α n e α 1 , … , e α n e^(alpha_(1)),dots,e^(alpha_(n))e^{\alpha_{1}}, \ldots, e^{\alpha_{n}}eα1,…,eαn are algebraically independent over Q Q Q\mathbb{Q}Q. This explains the terminology. The Ax-Lindemann principle in Example 4.2c) was proven by Pila [72] when S S SSS is a product Y ( 1 ) n × ( C ) k Y ( 1 ) n × C ∗ k Y(1)^(n)xx(C^(**))^(k)Y(1)^{n} \times\left(\mathbb{C}^{*}\right)^{k}Y(1)n×(C∗)k, by Ullmo-Yafaev [92] for projective Shimura varieties, by Pila-Tsimerman [76] for A g A g A_(g)\mathscr{A}_{g}Ag, and by Klingler-Ullmo-Yafaev [58] for any pure Shimura variety. The full Ax-Schanuel principle was proven by Mok-Pila-Tsimerman for pure Shimura varieties [64].
We conjectured in [55, coNJ. 7.5] that the Ax-Schanuel principle holds true for the bi-algebraic structure associated to a (graded-)polarized variation of (mixed) Z H S Z H S ZHS\mathbb{Z H S}ZHS on an arbitrary quasiprojective variety S S SSS. Bakker and Tsimerman proved this conjecture in the pure case:
Theorem 4.4 (Ax-Schanuel for Z V H S , [ 12 ] Z V H S , [ 12 ] ZVHS,[12]\mathbb{Z V H S ,}[12]ZVHS,[12] ). Let Φ : S an Γ D Φ : S an  → Γ ∖ D Phi:S^("an ")rarr Gamma\\D\Phi: S^{\text {an }} \rightarrow \Gamma \backslash DΦ:San →Γ∖D be a period map. Let V S × D ˇ V ⊂ S × D ˇ V sub S xxD^(ˇ)V \subset S \times \check{D}V⊂S×Dˇ be an algebraic subvariety. Let U U UUU be an irreducible complex analytic component of W ( S × Γ D D ) W ∩ S × Γ ∖ D D W nn(Sxx_(Gamma\\D)D)W \cap\left(S \times_{\Gamma \backslash D} D\right)W∩(S×Γ∖DD) such that
(4.1) codim S × D U < codim S × D ˇ W + codim S × D ( S × × Γ D D ) (4.1) codim S × D ⁡ U < codim S × D ˇ ⁡ W + codim S × D ⁡ S × × Γ ∖ D D {:(4.1)codim_(S xx D)U < codim_(S xxD^(ˇ))W+codim_(S xx D)(S xxxx_(Gamma\\D)D):}\begin{equation*} \operatorname{codim}_{S \times D} U<\operatorname{codim}_{S \times \check{D}} W+\operatorname{codim}_{S \times D}\left(S \times \times_{\Gamma \backslash D} D\right) \tag{4.1} \end{equation*}(4.1)codimS×D⁡U<codimS×Dˇ⁡W+codimS×D⁡(S××Γ∖DD)
Then the projection of U U UUU to S S SSS is contained in a strict weakly special subvariety of S S SSS for Φ Î¦ Phi\PhiΦ.
Remark 4.5. The results of [64] were extended by Gao [39] to mixed Shimura varieties of Kuga type. Recently the full Ax-Schanuel [55, coNJ. 7.5] for variations of mixed Hodge structures has been fully proven independently in [40] and [26].
The proof of Theorem 4.4 follows a strategy started in [58] and fully developed in [64] in the Shimura case, see [88] for an introduction. It does not use Theorem 3.14, but only a weak version equivalent to the Nilpotent Orbit Theorem, and relies crucially on the definable Chow Theorem 3.10, the Pila-Wilkie Theorem 3.8, and the proof that the volume (for the natural metric on Γ D ) Γ ∖ D ) Gamma\\D)\Gamma \backslash D)Γ∖D) of the intersection of a ball of radius R R RRR in Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D with the horizontal complex analytic subvariety Φ ( S an ) Φ S an  Phi(S^("an "))\Phi\left(S^{\text {an }}\right)Φ(San ) grows exponentially with R R RRR (a negative curvature property of the horizontal tangent bundle).

4.3. On the distribution of the Hodge locus

Theorem 4.4 is most useful, even in its simplest version of the Ax-Lindemann theorem. After Theorem 3.1 one would like to understand the distribution in S S SSS of the special subvarieties for V V V\mathbb{V}V. For instance, are there any geometric constraints on the Zariski closure of HL ( S , V ) HL ⁡ S , V ⊗ HL(S,V^(ox))\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right)HL⁡(S,V⊗) ? To approach this question, let us decompose the adjoint group G ad G ad  G^("ad ")\mathbf{G}^{\text {ad }}Gad  into a product G 1 × × G r G 1 × ⋯ × G r G_(1)xx cdots xxG_(r)\mathbf{G}_{1} \times \cdots \times \mathbf{G}_{r}G1×⋯×Gr of its simple factors. It gives rise (after passing to a finite étale covering if necessary) to a decomposition of the Hodge variety Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D into a product of Hodge varieties Γ 1 D 1 × × Γ r D r Γ 1 ∖ D 1 × ⋯ × Γ r ∖ D r Gamma_(1)\\D_(1)xx cdots xxGamma_(r)\\D_(r)\Gamma_{1} \backslash D_{1} \times \cdots \times \Gamma_{r} \backslash D_{r}Γ1∖D1×⋯×Γr∖Dr. A special subvariety Z Z ZZZ of S S SSS for V V V\mathbb{V}V is said of positive period dimension if dim C Φ ( Z a n ) > 0 dim C ⁡ Φ Z a n > 0 dim_(C)Phi(Z^(an)) > 0\operatorname{dim}_{\mathbb{C}} \Phi\left(Z^{\mathrm{an}}\right)>0dimC⁡Φ(Zan)>0; and of factorwise positive period dimension if, moreover, the projection of Φ ( Z a n ) Φ Z a n Phi(Z^(an))\Phi\left(Z^{\mathrm{an}}\right)Φ(Zan) on each factor Γ i D i Γ i ∖ D i Gamma_(i)\\D_(i)\Gamma_{i} \backslash D_{i}Γi∖Di has positive dimension. The Hodge locus of factorwise positive period dimension H L ( S , V ) f p o s H L S , V ⊗ f p o s HL(S,V^(ox))_(fpos)\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{fpos}}HL(S,V⊗)fpos is the union of the strict special subvarieties of positive period dimension, it is contained in the Hodge locus of positive period dimension H L ( S , V ) pos H L S , V ⊗ pos  HL(S,V^(ox))_("pos ")\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\text {pos }}HL(S,V⊗)pos  union of the strict special subvarieties of positive period dimension, and the two coincide if G ad G ad  G^("ad ")\mathbf{G}^{\text {ad }}Gad  is simple.
Using the Ax-Lindemann theorem special case of Theorem 4.4 and a global algebraicity result in the total bundle of V V V\mathcal{V}V, Otwinowska and the author proved the following:
Theorem 4.6 ([56]). Let V V V\mathbb{V}V be a polarized Z V H S Z V H S ZVHS\mathbb{Z} V H SZVHS on a smooth connected complex quasiprojective variety S S SSS. Then either HL ( S , V ) f p o s HL ⁡ S , V ⊗ f p o s HL (S,V^(ox))_(fpos)\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{fpos}}HL⁡(S,V⊗)fpos is Zariski-dense in S S SSS; or it is an algebraic subvariety of S S SSS (i.e., the set of strict special subvarieties of S S SSS for V V V\mathbb{V}V of factorwise positive period dimension has only finitely many maximal elements for the inclusion).
Example 4.7. The simplest example of Theorem 4.6 is the following. Let S A g S ⊂ A g S subA_(g)S \subset \mathscr{A}_{g}S⊂Ag be a Hodge-generic closed irreducible subvariety. Either the set of positive-dimensional closed
irreducible subvarieties of S S SSS which are not Hodge generic has finitely many maximal elements (for the inclusion), or their union is Zariski-dense in S S SSS.
Example 4.8. Let B P H 0 ( P C 3 , O ( d ) ) B ⊂ P H 0 P C 3 , O ( d ) B subPH^(0)(P_(C)^(3),O(d))B \subset \mathbb{P} H^{0}\left(\mathbb{P}_{\mathbb{C}}^{3}, \mathcal{O}(d)\right)B⊂PH0(PC3,O(d)) be the open subvariety parametrizing the smooth surfaces of degree d d ddd in P C 3 P C 3 P_(C)^(3)\mathbb{P}_{\mathbb{C}}^{3}PC3. Suppose d > 3 d > 3 d > 3d>3d>3. The classical Noether theorem states that any surface Y P C 3 Y ⊂ P C 3 Y subP_(C)^(3)Y \subset \mathbb{P}_{\mathbb{C}}^{3}Y⊂PC3 corresponding to a very general point [ Y ] B [ Y ] ∈ B [Y]in B[Y] \in B[Y]∈B has Picard group Z Z Z\mathbb{Z}Z : every curve on Y Y YYY is a complete intersection of Y Y YYY with another surface in P C 3 P C 3 P_(C)^(3)\mathbb{P}_{\mathbb{C}}^{3}PC3. The countable union N L ( B ) N L ( B ) NL(B)\mathrm{NL}(B)NL(B) of closed algebraic subvarieties of B B BBB corresponding to surfaces with bigger Picard group is called the Noether-Lefschetz locus of B B BBB. Let V B V → B Vrarr B\mathbb{V} \rightarrow BV→B be the Z V H S R 2 f Z prim Z V H S R 2 f ∗ Z prim  ZVHSR^(2)f_(**)Z_("prim ")\mathbb{Z V H S} R^{2} f_{*} \mathbb{Z}_{\text {prim }}ZVHSR2f∗Zprim , where f : y B f : y → B f:y rarr Bf: y \rightarrow Bf:y→B denotes the universal family of surfaces of degree d d ddd. Clearly N L ( B ) N L ( B ) ⊂ NL(B)sub\mathrm{NL}(B) \subsetNL(B)⊂ HL ( B , V ) HL ⁡ B , V ⊗ HL(B,V^(ox))\operatorname{HL}\left(B, \mathbb{V}^{\otimes}\right)HL⁡(B,V⊗). Green (see [94, PRoP. 5.20]) proved that NL ( B ) NL ⁡ ( B ) NL(B)\operatorname{NL}(B)NL⁡(B), hence also HL ( B , V ) HL ⁡ B , V ⊗ HL(B,V^(ox))\operatorname{HL}\left(B, \mathbb{V}^{\otimes}\right)HL⁡(B,V⊗), is analytically dense in B B BBB. Now Theorem 4.6 implies the following: Let S B S ⊂ B S sub BS \subset BS⊂B be a Hodgegeneric closed irreducible subvariety. Either S HL ( B , V ) fpos S ∩ HL ⁡ B , V ⊗ fpos  S nn HL (B,V^(ox))_("fpos ")S \cap \operatorname{HL}\left(B, \mathbb{V}^{\otimes}\right)_{\text {fpos }}S∩HL⁡(B,V⊗)fpos  contains only finitely many maximal positive-dimensional closed irreducible subvarieties of S S SSS, or the union of such subvarieties is Zariski-dense in S S SSS.

5. TYPICAL AND ATYPICAL INTERSECTIONS: THE ZILBER-PINK CONJECTURE FOR PERIOD MAPS

5.1. The Zilber-Pink conjecture for Z V H S Z V H S ZVHS\mathbb{Z V H S}ZVHS : Conjectures

In the same way that the Ax-Schanuel principle controls the geometry of bialgebraic structures, the diophantine geometry of Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯-bialgebraic structures is controlled by the following heuristic:
Atypical intersection principle. Let S S SSS be an irreducible algebraic Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯-variety endowed with a Q ¯ a Q ¯ a bar(Q)a \overline{\mathbb{Q}}aQ¯-bialgebraic structure. Then the union S atyp S atyp  S_("atyp ")S_{\text {atyp }}Satyp  of atypical Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯-bialgebraic subvarieties of S S SSS is an algebraic subvariety of S S SSS (i.e., it contains only finitely many atypical Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯-bialgebraic subvarieties maximal for the inclusion).
Here a Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯-bialgebraic subvariety Y S Y ⊂ S Y sub SY \subset SY⊂S is said to be atypical for the given bialgebraic structure on S S SSS if it is obtained as an excess intersection of f ( S an ~ ) f S an  ~ f(( widetilde(S^("an "))))f\left(\widetilde{S^{\text {an }}}\right)f(San ~) with its model f ( Y ~ ) ¯ f ( Y ~ ) ¯ bar(f(( tilde(Y))))\overline{f(\tilde{Y})}f(Y~)¯ Zar Z ⊂ Z sub Z\subset Z⊂Z; and S atyp S atyp  S_("atyp ")S_{\text {atyp }}Satyp  denotes the union of all atypical subvarieties of S S SSS. As a particular case of the atypical intersection principle:
Sparsity of arithmetic points principle. Let S S SSS be an irreducible algebraic Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯-variety endowed with a Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯-bialgebraic structure. Then any irreducible algebraic subvariety of S S SSS containing a Zariski-dense set of atypical arithmetic points is a Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯-bialgebraic subvariety.
This principle that arithmetic points are sparse is a theorem of Mann [63] in Example 4.2a). For abelian varieties over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯ (Example 4.2b)), this is the Manin-Mumford conjecture proven first by Raynaud [80], saying that an irreducible subvariety of an abelian variety over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯ containing a Zariski-dense set of torsion point is the translate of an abelian subvariety by a torsion point. For Shimura varieties of abelian type (Example 4.2c)), this is the classical André-Oort conjecture [1,67] stating that an irreducible subvariety of a Shimura variety containing a Zariski-dense set of special points is special. It has been proven in this
case using tame geometry and following the strategy proposed by Pila-Zannier [78] (let us mention [ 3 , 58 , 72 , 76 , 87 , 89 , 98 ] [ 3 , 58 , 72 , 76 , 87 , 89 , 98 ] [3,58,72,76,87,89,98][3,58,72,76,87,89,98][3,58,72,76,87,89,98]; and [38] in the mixed case; see [59] for a survey). Recently the André-Oort conjecture in full generality has been obtained in [75], reducing to the case of abelian type using ingredients from p p ppp-adic Hodge theory. We refer to [99] for many examples of atypical intersection problems.
In the case of Shimura varieties (Example 4.2c)) the general atypical intersection principle is the Zilber-Pink conjecture [51,79,100]. Only very few instances of the Zilber-Pink conjecture are known outside of the André-Oort conjecture, see [ 27 , 49 , 50 ] [ 27 , 49 , 50 ] [27,49,50][27,49,50][27,49,50], for example.
For a general polarized Z V H S V Z V H S V ZVHSV\mathbb{Z V H S} \mathbb{V}ZVHSV with period map Φ : S an Γ D Φ : S an  → Γ ∖ D Phi:S^("an ")rarr Gamma\\D\Phi: S^{\text {an }} \rightarrow \Gamma \backslash DΦ:San →Γ∖D, which we can assume to be proper without loss of generality, we already mentioned that even the geometric characterization of the Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯-bialgebraic subvarieties as the special subvarieties is unknown. Replacing the Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯-bialgebraic subvarieties of S S SSS by the special ones, we define:
Definition 5.1. A special subvariety Z = Φ 1 ( Γ Z D Z ) 0 S Z = Φ − 1 Γ Z ∖ D Z 0 ⊂ S Z=Phi^(-1)(Gamma_(Z)\\D_(Z))^(0)sub SZ=\Phi^{-1}\left(\Gamma_{Z} \backslash D_{Z}\right)^{0} \subset SZ=Φ−1(ΓZ∖DZ)0⊂S is said atypical if either Z Z ZZZ is singular for V V V\mathbb{V}V (meaning that Φ ( Z a n ) Φ Z a n Phi(Z^(an))\Phi\left(Z^{\mathrm{an}}\right)Φ(Zan) is contained in the singular locus of the complex analytic variety Φ ( S a n ) Φ S a n Phi(S^(an))\Phi\left(S^{\mathrm{an}}\right)Φ(San) ), or if Φ ( S a n ) Φ S a n Phi(S^(an))\Phi\left(S^{\mathrm{an}}\right)Φ(San) and Γ Z D Z Γ Z ∖ D Z Gamma_(Z)\\D_(Z)\Gamma_{Z} \backslash D_{Z}ΓZ∖DZ do not intersect generically along Φ ( Z ) Φ ( Z ) Phi(Z)\Phi(Z)Φ(Z) :
codim Γ D Φ ( Z an ) < codim Γ D Φ ( S a n ) + codim Γ D Γ Z D Z codim Γ ∖ D ⁡ Φ Z an  < codim Γ ∖ D ⁡ Φ S a n + codim Γ ∖ D ⁡ Γ Z ∖ D Z codim_(Gamma\\D)Phi(Z^("an ")) < codim_(Gamma\\D)Phi(S^(an))+codim_(Gamma\\D)Gamma_(Z)\\D_(Z)\operatorname{codim}_{\Gamma \backslash D} \Phi\left(Z^{\text {an }}\right)<\operatorname{codim}_{\Gamma \backslash D} \Phi\left(S^{\mathrm{an}}\right)+\operatorname{codim}_{\Gamma \backslash D} \Gamma_{Z} \backslash D_{Z}codimΓ∖D⁡Φ(Zan )<codimΓ∖D⁡Φ(San)+codimΓ∖D⁡ΓZ∖DZ
Otherwise, it is said to be typical.
Defining the atypical Hodge locus H L ( S , V ) atyp H L ( S , V ) H L S , V ⊗ atyp  ⊂ H L S , V ⊗ HL(S,V^(ox))_("atyp ")subHL(S,V^(ox))\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\text {atyp }} \subset \mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)HL(S,V⊗)atyp ⊂HL(S,V⊗) as the union of the atypical special subvarieties of S S SSS for V V V\mathbb{V}V, we obtain the following precise atypical intersection principle for Z V H S Z V H S ZVHS\mathbb{Z V H S}ZVHS, first proposed in [55] in a more restrictive form:
Conjecture 5.2 (Zilber-Pink conjecture for Z V H S , [ 13 , 55 ] ) Z V H S , [ 13 , 55 ] ) ZVHS,[13,55])\mathbb{Z V H S},[13,55])ZVHS,[13,55]). Let V V V\mathbb{V}V be a polarizable Z V H S Z V H S ZVHS\mathbb{Z} V H SZVHS on an irreducible smooth quasiprojective variety S S SSS. The atypical Hodge locus H L ( S , V ) atyp H L S , V ⊗ atyp  HL(S,V^(ox))_("atyp ")\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\text {atyp }}HL(S,V⊗)atyp  is a finite union of atypical special subvarieties of S S SSS for V V V\mathbb{V}V. Equivalently, the set of atypical special subvarieties of S S SSS for V V V\mathbb{V}V has finitely many maximal elements for the inclusion.
Notice that this conjecture is in some sense more general than the above atypical intersection principle, as we do not assume that S S SSS is defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯; this has to be compared to the fact that the Manin-Mumford conjecture holds true for every complex abelian variety, not necessarily defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯.
Example 5.3. Recently Baldi and Ullmo [14] proved a special case of Conjecture 5.2 of much interest. Margulis' arithmeticity theorem states that any lattice in a simple real Lie group G G GGG of real rank at least 2 is arithmetic: it is commensurable with a group G ( Z ) G ( Z ) G(Z)\mathbf{G}(\mathbb{Z})G(Z), for G G G\mathbf{G}G a Q Q Q\mathbb{Q}Q algebraic group such that G ( R ) = G G ( R ) = G G(R)=G\mathbf{G}(\mathbb{R})=GG(R)=G up to a compact factor. On the other hand, the structure of lattices in a simple real Lie group of rank 1, like the group PU ( n , 1 ) PU ⁡ ( n , 1 ) PU(n,1)\operatorname{PU}(n, 1)PU⁡(n,1) of holomorphic isometries of the complex unit ball B C n B C n B_(C)^(n)\mathbf{B}_{\mathbb{C}}^{n}BCn endowed with its Bergman metric, is an essentially open question. In particular, there exist nonarithmetic lattices in PU ( n , 1 ) , n = 2 , 3 PU ⁡ ( n , 1 ) , n = 2 , 3 PU(n,1),n=2,3\operatorname{PU}(n, 1), n=2,3PU⁡(n,1),n=2,3. Let ι : Λ PU ( n , 1 ) ι : Λ ↪ PU ⁡ ( n , 1 ) iota:Lambda↪PU(n,1)\iota: \Lambda \hookrightarrow \operatorname{PU}(n, 1)ι:Λ↪PU⁡(n,1) be a lattice. The ball quotient S a n := Λ B C n S a n := Λ ∖ B C n S^(an):=Lambda\\B_(C)^(n)S^{\mathrm{an}}:=\Lambda \backslash \mathbf{B}_{\mathbb{C}}^{n}San:=Λ∖BCn is the analytification of a complex algebraic variety S S SSS. By results of Simpson and Esnault-Groechenig, there exists a Z V H S Φ : S an Γ ( B C n × D ) Z V H S Φ : S an  → Γ ∖ B C n × D ′ ZVHSPhi:S^("an ")rarr Gamma\\(B_(C)^(n)xxD^('))\mathbb{Z V H S} \Phi: S^{\text {an }} \rightarrow \Gamma \backslash\left(\mathbf{B}_{\mathbb{C}}^{n} \times D^{\prime}\right)ZVHSΦ:San →Γ∖(BCn×D′) with monodromy representation ρ : Λ P U ( n , 1 ) × G ρ : Λ → P U ( n , 1 ) × G ′ rho:Lambda rarrPU(n,1)xxG^(')\rho: \Lambda \rightarrow \mathrm{PU}(n, 1) \times G^{\prime}ρ:Λ→PU(n,1)×G′
whose first factor Λ P U ( n , 1 ) Λ → P U ( n , 1 ) Lambda rarrPU(n,1)\Lambda \rightarrow \mathrm{PU}(n, 1)Λ→PU(n,1) is the rigid representation ι ι iota\iotaι. The special subvarieties of S S SSS for V V V\mathbb{V}V are the totally geodesic complex subvarieties of S an S an  S^("an ")S^{\text {an }}San . When Λ Î› Lambda\LambdaΛ is nonarithmetic, they are automatically atypical. In accordance with Conjecture 5.2 in this case, Baldi and Ullmo prove that if Λ Î› Lambda\LambdaΛ is nonarithmetic, then S an S an  S^("an ")S^{\text {an }}San  contains only finitely many maximal totally geodesic subvarieties. This result has been proved independently by Bader, Fisher, Miller, and Stover [7], using completely different methods from homogeneous dynamics.
Among the special points for a Z V H S V Z V H S V ZVHSV\mathbb{Z V H S} \mathbb{V}ZVHSV, the C M C M CM\mathrm{CM}CM-points (i.e., those for which the Mumford-Tate group is a torus) are always atypical except if the generic Hodge datum ( G , D ) ( G , D ) (G,D)(\mathbf{G}, D)(G,D) is of Shimura type and the period map Φ Î¦ Phi\PhiΦ is dominant. Hence, as explained in [55, SECTION 5.2], Conjecture 5.2 implies the following:
Conjecture 5.4 (André-Oort conjecture for Z V H S Z V H S ZVHS\mathbb{Z V H S}ZVHS, [55]). Let V V V\mathbb{V}V be a polarizable Z V H S Z V H S ZVHS\mathbb{Z} V H SZVHS on an irreducible smooth quasiprojective variety S. If S S SSS contains a Zariski-dense set of CMpoints then the generic Hodge datum ( G , D ) ( G , D ) (G,D)(\mathbf{G}, D)(G,D) of V V V\mathbb{V}V is a Shimura datum, and the period map Φ : S a n Γ D Φ : S a n → Γ ∖ D Phi:S^(an)rarr Gamma\\D\Phi: S^{\mathrm{an}} \rightarrow \Gamma \backslash DΦ:San→Γ∖D is an algebraic map, dominant on the Shimura variety Γ D Γ ∖ D Gamma\\D\Gamma \backslash DΓ∖D.
Example 5.5. Consider the Calabi-Yau Hodge structure V V VVV of weight 3 with Hodge numbers h 3 , 0 = h 2 , 1 = 1 h 3 , 0 = h 2 , 1 = 1 h^(3,0)=h^(2,1)=1h^{3,0}=h^{2,1}=1h3,0=h2,1=1 given by the mirror dual quintic. Its universal deformation space S S SSS is the projective line minus 3 points, which carries a Z V H S V Z V H S V ZVHSV\mathbb{Z V H S} \mathbb{V}ZVHSV of the same type. This gives a nontrivial period map Φ : S an Γ D Φ : S an  → Γ ∖ D Phi:S^("an ")rarr Gamma\\D\Phi: S^{\text {an }} \rightarrow \Gamma \backslash DΦ:San →Γ∖D, where D = S p ( 4 , R ) / U ( 1 ) × U ( 1 ) D = S p ( 4 , R ) / U ( 1 ) × U ( 1 ) D=Sp(4,R)//U(1)xx U(1)D=\mathbf{S p}(4, \mathbb{R}) / U(1) \times U(1)D=Sp(4,R)/U(1)×U(1) is a 4-dimensional period domain. This period map is known not to factorize through a Shimura subvariety (its algebraic monodromy group is S p 4 S p 4 Sp4\mathbf{S p} 4Sp4 ). Conjecture 5.4 in that case predicts that S S SSS contains only finitely many points CM-points s s sss. A version of this prediction already appears in [48]. The more general Conjecture 5.2 also predicts that S S SSS contains only finitely many points s s sss where V s V s V_(s)\mathbb{V}_{s}Vs splits as a direct sum of two (Tate twisted) weight one Hodge structures ( V s 2 , 1 V s 1 , 2 ) V s 2 , 1 ⊕ V s 1 , 2 (V_(s)^(2,1)o+V_(s)^(1,2))\left(\mathbb{V}_{s}^{2,1} \oplus \mathbb{V}_{s}^{1,2}\right)(Vs2,1⊕Vs1,2) and its orthogonal for the Hodge metric ( V s 3 , 0 V s 0 , 3 ) V s 3 , 0 ⊕ V s 0 , 3 (V_(s)^(3,0)o+V_(s)^(0,3))\left(\mathbb{V}_{s}^{3,0} \oplus \mathbb{V}_{s}^{0,3}\right)(Vs3,0⊕Vs0,3) (the so-called "rank two attractors" points, see [66]).
Conjecture 5.2 about the atypical Hodge locus takes all its meaning if we compare it to the expected behavior of its complement, the typical Hodge locus HL ( S , V ) t y p := HL ⁡ S , V ⊗ t y p := HL (S,V^(ox))_(typ):=\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{typ}}:=HL⁡(S,V⊗)typ:= HL ( S , V ) H L ( S , V atyp ) : HL ⁡ S , V ⊗ ∖ H L S , V atyp  ⊗ : HL(S,V^(ox))\\HL(S,V_("atyp ")^(ox)):\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right) \backslash \mathrm{HL}\left(S, \mathbb{V}_{\text {atyp }}^{\otimes}\right):HL⁡(S,V⊗)∖HL(S,Vatyp ⊗):
Conjecture 5.6 (Density of the typical Hodge locus, [13]). If H L ( S , V ) t y p H L S , V ⊗ t y p HL(S,V^(ox))_(typ)\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{typ}}HL(S,V⊗)typ is not empty then it is dense (for the analytic topology) in S a n S a n S^(an)S^{\mathrm{an}}San.
Conjectures 5.2 and 5.6 imply immediately the following, which clarifies the possible alternatives in Theorem 4.6:
Conjecture 5.7 ([13]). Let V V V\mathbb{V}V be a polarizable Z V H S Z V H S ZVHS\mathbb{Z} V H SZVHS on an irreducible smooth quasiprojective variety S S SSS. If H L ( S , V ) t y p H L S , V ⊗ t y p HL(S,V^(ox))_(typ)\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{typ}}HL(S,V⊗)typ is empty then H L ( S , V ) H L S , V ⊗ HL(S,V^(ox))\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)HL(S,V⊗) is algebraic; otherwise, H L ( S , V ) H L S , V ⊗ HL(S,V^(ox))\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)HL(S,V⊗) is analytically dense in S an S an  S^("an ")S^{\text {an }}San .

5.2. The Zilber-Pink conjecture for Z Z Z\mathbb{Z}Z VHS: Results

In [13] Baldi, Ullmo, and I establish the geometric part of Conjecture 5.2: the maximal atypical special subvarieties of positive period dimension arise in a finite number of families whose geometry is well understood. We cannot say anything on the atypical locus of zero period dimension (for which different ideas are certainly needed):
Theorem 5.8 (Geometric Zilber-Pink, [13]). Let V V V\mathbb{V}V be a polarizable Z V H S Z V H S ZVHS\mathbb{Z} V H SZVHS on a smooth connected complex quasiprojective variety S S SSS. Let Z Z ZZZ be an irreducible component of the Zariski closure of H L ( S , V ) p o s , atyp := H L ( S , V ) p o s H L ( S , V ) atyp H L S , V ⊗ p o s ,  atyp  := H L S , V ⊗ p o s ∩ H L S , V ⊗ atyp  HL(S,V^(ox))_(pos," atyp "):=HL(S,V^(ox))_(pos)nnHL(S,V^(ox))_("atyp ")\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{pos}, \text { atyp }}:=\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{pos}} \cap \mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\text {atyp }}HL(S,V⊗)pos, atyp :=HL(S,V⊗)pos∩HL(S,V⊗)atyp  in S S SSS. Then:
(a) Either Z Z ZZZ is a maximal atypical special subvariety;
(b) Or the generic adjoint Hodge datum ( G Z ad , D G Z G Z ad  , D G Z G_(Z)^("ad "),D_(G_(Z))\mathbf{G}_{Z}^{\text {ad }}, D_{G_{Z}}GZad ,DGZ ) decomposes as a nontrivial product ( G , D ) × ( G , D ) G ′ , D ′ × G ′ ′ , D ′ ′ (G^('),D^('))xx(G^(''),D^(''))\left(\mathbf{G}^{\prime}, D^{\prime}\right) \times\left(\mathbf{G}^{\prime \prime}, D^{\prime \prime}\right)(G′,D′)×(G′′,D′′), inducing (after replacing S S SSS by a finite étale cover if necessary)
Φ Z a n = ( Φ , Φ ) : Z a n Γ G Z D G Z = Γ D × Γ D Γ D Φ ∣ Z a n = Φ ′ , Φ ′ ′ : Z a n → Γ G Z ∖ D G Z = Γ ′ ∖ D ′ × Γ ′ ′ ∖ D ′ ′ ⊂ Γ ∖ D Phi_(∣Z^(an))=(Phi^('),Phi^('')):Z^(an)rarrGamma_(G_(Z))\\D_(G_(Z))=Gamma^(')\\D^(')xxGamma^('')\\D^('')sub Gamma\\D\Phi_{\mid Z^{\mathrm{an}}}=\left(\Phi^{\prime}, \Phi^{\prime \prime}\right): Z^{\mathrm{an}} \rightarrow \Gamma_{\mathbf{G}_{Z}} \backslash D_{G_{Z}}=\Gamma^{\prime} \backslash D^{\prime} \times \Gamma^{\prime \prime} \backslash D^{\prime \prime} \subset \Gamma \backslash DΦ∣Zan=(Φ′,Φ′′):Zan→ΓGZ∖DGZ=Γ′∖D′×Γ′′∖D′′⊂Γ∖D
such that Z contains a Zariski-dense set of atypical special subvarieties for Φ Î¦ ′ ′ Phi^('')\Phi^{\prime \prime}Φ′′ of zero period dimension. Moreover, Z Z ZZZ is Hodge generic in the special subvariety Φ 1 ( Γ G Z D G Z ) 0 Φ − 1 Γ G Z ∖ D G Z 0 Phi^(-1)(Gamma_(G_(Z))\\D_(G_(Z)))^(0)\Phi^{-1}\left(\Gamma_{\mathbf{G}_{Z}} \backslash D_{G_{Z}}\right)^{0}Φ−1(ΓGZ∖DGZ)0 of S S SSS for Φ Î¦ Phi\PhiΦ, which is typical.
Conjecture 5.2, which also takes into account the atypical special subvarieties of zero period dimension, predicts that the branch (b) of the alternative in the conclusion of Theorem 5.8 never occurs. Theorem 5.8 is proven using properties of definable sets and the Ax-Schanuel Theorem 4.4, following an idea originating in [89].
As an application of Theorem 5.8, let us consider the Shimura locus of S S SSS for V V V\mathbb{V}V, namely the union of the special subvarieties of S S SSS for V V V\mathbb{V}V which are of Shimura type (but not necessarily with dominant period maps). In [55], I asked (generalizing the André-Oort conjecture for Z V H S Z V H S ZVHS\mathbb{Z V H S}ZVHS ) whether a polarizable Z V H S V Z V H S V ZVHSV\mathbb{Z V H S} \mathbb{V}ZVHSV on S S SSS whose Shimura locus in Zariskidense in S S SSS is necessarily of Shimura type. As a corollary of Theorem 5.8 we obtain:
Theorem 5.9 ([13]). Let V V V\mathbb{V}V be a polarizable Z V H S Z V H S ZVHS\mathbb{Z} V H SZVHS on a smooth irreducible complex quasiprojective variety S S SSS, with generic Hodge datum ( G , D ) ( G , D ) (G,D)(\mathbf{G}, D)(G,D). Suppose that the Shimura locus of S S SSS for V V V\mathbb{V}V of positive period dimension is Zariski-dense in S S SSS. If G ad G ad  G^("ad ")\mathbf{G}^{\text {ad }}Gad  is simple then V V V\mathbb{V}V is of Shimura type.

5.3. On the algebraicity of the Hodge locus

In view of Conjecture 5.7, it is natural to ask if there a simple combinatorial criterion on ( G , D ) ( G , D ) (G,D)(\mathbf{G}, D)(G,D) for deciding whether HL ( S , V ) typ HL ⁡ ( S , V ) typ  HL(S,V)_("typ ")\operatorname{HL}(S, \mathbb{V})_{\text {typ }}HL⁡(S,V)typ  is empty. Intuitively, one expects that the more "complicated" the Hodge structure is, the smaller the typical Hodge locus should be, due to the constraint imposed by Griffiths' transversality. Let us measure the complexity of
V V V\mathbb{V}V by its level: when G ad G ad  G^("ad ")\mathbf{G}^{\text {ad }}Gad  is simple, it is the greatest integer k k kkk such that g k , k 0 g k , − k ≠ 0 g^(k,-k)!=0\mathrm{g}^{k,-k} \neq 0gk,−k≠0 in the Hodge decomposition of the Lie algebra g g ggg of G G G\mathbf{G}G; in general one takes the minimum of these integers obtained for each simple Q Q Q\mathbb{Q}Q-factor of G ad G ad  G^("ad ")\mathbf{G}^{\text {ad }}Gad . While strict typical special subvarieties
usually abound for Z V H S s Z V H S s ZVHSs\mathbb{Z V H S s}ZVHSs of level one (e.g., families of abelian varieties, see Example 4.7; or families of K3 surfaces) and can occur in level two (see Example 4.8), they do not exist in level at least three!
Theorem 5.10 ([13]). Let V V V\mathbb{V}V be a polarizable Z V H S Z V H S ZVHS\mathbb{Z} V H SZVHS on a smooth connected complex quasiprojective variety S S SSS. If V V V\mathbb{V}V is of level at least 3 then HL ( S , V ) t y p = HL ⁡ S , V ⊗ t y p = ∅ HL (S,V^(ox))_(typ)=O/\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{typ}}=\emptysetHL⁡(S,V⊗)typ=∅ (and thus H L ( S , V ) = H L S , V ⊗ = HL(S,V^(ox))=\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)=HL(S,V⊗)= HL ( S , V ) atyp ) HL ⁡ S , V ⊗ atyp  {: HL (S,V^(ox))_("atyp "))\left.\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\text {atyp }}\right)HL⁡(S,V⊗)atyp ).
The proof of Theorem 5.10 is purely Lie-theoretic. Let ( G , D ) ( G , D ) (G,D)(\mathbf{G}, D)(G,D) be the generic Hodge datum of V V V\mathbb{V}V and Φ : S an Γ D Φ : S an  → Γ ∖ D Phi:S^("an ")rarr Gamma\\D\Phi: S^{\text {an }} \rightarrow \Gamma \backslash DΦ:San →Γ∖D its period map. Suppose that Y S Y ⊂ S Y sub SY \subset SY⊂S is a typical special subvariety, with generic Hodge datum ( G Y , D Y G Y , D Y G_(Y),D_(Y)\mathbf{G}_{Y}, D_{Y}GY,DY ). The typicality condition and the horizontality of the period map Φ Î¦ Phi\PhiΦ imply that g Y i , i = g i , i g Y − i , i = g − i , i g_(Y)^(-i,i)=g^(-i,i)\mathrm{g}_{Y}^{-i, i}=\mathrm{g}^{-i, i}gY−i,i=g−i,i for all i 2 i ≥ 2 i >= 2i \geq 2i≥2 (for the Hodge structures on the Lie algebras g Y g Y g_(Y)\mathrm{g}_{Y}gY and g g g\mathrm{g}g defined by some point of D Y D Y D_(Y)D_{Y}DY ). Under the assumption that V V V\mathbb{V}V has level at least 3, we show that this is enough to ensure that g Y = g g Y = g g_(Y)=g\mathrm{g}_{Y}=\mathrm{g}gY=g, hence Y = S Y = S Y=SY=SY=S. Hence there are no strict typical special subvariety.
Notice that Conjecture 5.2 and Theorem 5.10 imply:
Conjecture 5.11 (Algebraicity of the Hodge locus in level at least 3, [13]). Let V V V\mathbb{V}V be a polarizable Z V H S Z V H S ZVHS\mathbb{Z} V H SZVHS on a smooth connected complex quasiprojective variety S S SSS. If V V V\mathbb{V}V is of level at least 3 then H L ( S , V ) H L S , V ⊗ HL(S,V^(ox))\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)HL(S,V⊗) is algebraic.
The main result of [13], which follows immediately from Theorems 5.8 and 5.10, is the following stunning geometric reinforcement of Theorems 3.1 and 4.6:
Theorem 5.12 ([13]). If V V V\mathbb{V}V is of level at least 3 then HL ( S , V ) f p o s HL ⁡ S , V ⊗ f p o s HL (S,V^(ox))_(fpos)\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{fpos}}HL⁡(S,V⊗)fpos is algebraic.
As a simple geometric illustration of Theorem 5.12, we prove the following, to be contrasted with the n = 2 n = 2 n=2n=2n=2 case (see Example 4.8):
Corollary 5.13. Let P C N ( n , d ) P C N ( n , d ) P_(C)^(N(n,d))\mathbf{P}_{\mathbb{C}}^{N(n, d)}PCN(n,d) be the projective space parametrizing the hypersurfaces X X XXX of P C n + 1 P C n + 1 P_(C)^(n+1)\mathbf{P}_{\mathbb{C}}^{n+1}PCn+1 of degree d (where N ( n , d ) = ( n + d + 1 d ) 1 N ( n , d ) = ( n + d + 1 d ) − 1 N(n,d)=((n+d+1)/(d))-1N(n, d)=\binom{n+d+1}{d}-1N(n,d)=(n+d+1d)−1 ). Let U n , d P C N ( n , d ) U n , d ⊂ P C N ( n , d ) U_(n,d)subP_(C)^(N(n,d))U_{n, d} \subset \mathbf{P}_{\mathbb{C}}^{N(n, d)}Un,d⊂PCN(n,d) be the Zariskiopen subset parametrizing the smooth hypersurfaces X X XXX and let V U n , d V → U n , d VrarrU_(n,d)\mathbb{V} \rightarrow U_{n, d}V→Un,d be the Z V H S Z V H S ZVHS\mathbb{Z} V H SZVHS corresponding to the primitive cohomology H n ( X , Z ) prim. H n ( X , Z ) prim.  H^(n)(X,Z)_("prim. ")H^{n}(X, \mathbb{Z})_{\text {prim. }}Hn(X,Z)prim. . If n 3 n ≥ 3 n >= 3n \geq 3n≥3 and d > 5 d > 5 d > 5d>5d>5, then HL ( U n , d , V ) p o s U n , d HL ⁡ U n , d , V ⊗ p o s ⊂ U n , d HL (U_(n,d),V^(ox))_(pos)subU_(n,d)\operatorname{HL}\left(U_{n, d}, \mathbb{V}^{\otimes}\right)_{\mathrm{pos}} \subset U_{n, d}HL⁡(Un,d,V⊗)pos⊂Un,d is algebraic.

5.4. On the typical Hodge locus in level one and two

In the direction of Conjecture 5.6, we obtain:
Theorem 5.14 (Density of the typical locus, [13]). Let V V V\mathbb{V}V be a polarized Z V H S Z V H S ZVHS\mathbb{Z} V H SZVHS on a smooth connected complex quasiprojective variety S. If the typical Hodge locus H L ( S , V ) t y p H L S , V ⊗ t y p HL(S,V^(ox))_(typ)\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{typ}}HL(S,V⊗)typ is nonempty (hence the level of V V V\mathbb{V}V is one or two by Theorem 5.10) then HL ( S , V ) HL ⁡ S , V ⊗ HL(S,V^(ox))\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right)HL⁡(S,V⊗) is analytically (hence Zariski) dense in S S SSS.
Notice that, in Theorem 5.14, we also treat the typical Hodge locus of zero period dimension. Theorem 5.14 is new even for S S SSS a subvariety of a Shimura variety. Its proof is inspired by the arguments of Chai [24] in that case.
It remains to find a criterion for deciding whether, in level one or two, the typical Hodge locus HL ( S , V ) typ HL ⁡ S , V ⊗ typ  HL (S,V^(ox))_("typ ")\operatorname{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\text {typ }}HL⁡(S,V⊗)typ  is empty or not. We refer to [57, THEOREM 2.15] and [85, 86] for results in this direction.

6. ARITHMETIC ASPECTS

We turn briefly to some arithmetic aspects of period maps.

6.1. Field of definition of special subvarieties

Once more the geometric case provides us with a motivation and a heuristic. Let f : X S f : X → S f:X rarr Sf: X \rightarrow Sf:X→S be a smooth projective morphism of connected algebraic varieties defined over a number field L C L ⊂ C L subCL \subset \mathbb{C}L⊂C and let V V V\mathbb{V}V be the natural polarizable Z V H S Z V H S ZVHS\mathbb{Z V H S}ZVHS on S an S an  S^("an ")S^{\text {an }}San  with underlying local system R f an Z R ∙ f ∗ an  Z R^(∙)f_(**)^("an ")ZR^{\bullet} f_{*}^{\text {an }} \mathbb{Z}R∙f∗an Z. In that case, the Hodge conjecture implies that each special subvariety Y Y YYY of S S SSS for V V V\mathbb{V}V is defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯ and that each of the Gal ( Q ¯ / L ) Gal ⁡ ( Q ¯ / L ) Gal( bar(Q)//L)\operatorname{Gal}(\overline{\mathbb{Q}} / L)Gal⁡(Q¯/L)-conjugates of Y Y YYY is again a special subvariety of S S SSS for V V V\mathbb{V}V. More generally, let us say that a polarized Z V H S V = Z V H S V = ZVHSV=\mathbb{Z} V H S \mathbb{V}=ZVHSV= ( V Z , ( V , F , ) , q ) V Z , V , F ∙ , ∇ , q (V_(Z),(V,F^(∙),grad),q)\left(\mathbb{V}_{\mathbb{Z}},\left(\mathcal{V}, F^{\bullet}, \nabla\right), \mathrm{q}\right)(VZ,(V,F∙,∇),q) on S a n S a n S^(an)S^{\mathrm{an}}San is defined over a number field L C L ⊂ C L subCL \subset \mathbb{C}L⊂C if S , V , F S , V , F ∙ S,V,F^(∙)S, \mathcal{V}, F^{\bullet}S,V,F∙ and ∇ grad\nabla∇ are defined over L L LLL (with the obvious compatibilities).
Conjecture 6.1. Let V V V\mathbb{V}V be a Z V H S Z V H S ZVHS\mathbb{Z} V H SZVHS defined over a number field L C L ⊂ C L subCL \subset \mathbb{C}L⊂C. Then any special subvariety of S S SSS for V V V\mathbb{V}V is defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯, and any of its finitely many G a l ( Q ¯ / L ) G a l ( Q ¯ / L ) Gal( bar(Q)//L)\mathrm{Gal}(\overline{\mathbb{Q}} / L)Gal(Q¯/L)-conjugates is a special subvariety of S S SSS for V V V\mathbb{V}V.
There are only few results in that direction: see [95, THEOREM 0.6] for a proof under a strong geometric assumption; and [81], where it is shown that when S S SSS (not necessarily V V V\mathbb{V}V ) is defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯, then a special subvariety of S S SSS for V V V\mathbb{V}V is defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯ if and only if it contains a Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯-point of S S SSS. In [57] Otwinowska, Urbanik, and I provide a simple geometric criterion for a special subvariety of S S SSS for V V V\mathbb{V}V to satisfy Conjecture 6.1. In particular we obtain:
Theorem 6.2 ([57]). Let V V V\mathbb{V}V be a polarized Z V H S Z V H S ZVHS\mathbb{Z} V H SZVHS on a smooth connected complex quasiprojective variety S. Suppose that the adjoint generic Mumford-Tate group G ad G ad  G^("ad ")\mathbf{G}^{\text {ad }}Gad  of V V V\mathbb{V}V is simple. If S S SSS is defined over a number field L L LLL, then any maximal (strict) special subvariety Y S Y ⊂ S Y sub SY \subset SY⊂S of positive period dimension is defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯. If, moreover, V V V\mathbb{V}V is defined over L L LLL then the finitely many Gal ( Q ¯ / L ) Gal ⁡ ( Q ¯ / L ) Gal( bar(Q)//L)\operatorname{Gal}(\overline{\mathbb{Q}} / L)Gal⁡(Q¯/L)-translates of Y Y YYY are special subvarieties of S S SSS for V V V\mathbb{V}V.
As a corollary of Theorems 5.12 and 6.2, one obtains the following, which applies for instance in the situation of Corollary 5.13.
Corollary 6.3. Let V V V\mathbb{V}V be a polarized variation of Z Z Z\mathbb{Z}Z-Hodge structure on a smooth connected quasiprojective variety S S SSS. Suppose that V V V\mathbb{V}V is of level at least 3 , and that it is defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯. Then H L ( S , V ) f p o s H L S , V ⊗ f p o s HL(S,V^(ox))_(fpos)\mathrm{HL}\left(S, \mathbb{V}^{\otimes}\right)_{\mathrm{fpos}}HL(S,V⊗)fpos is an algebraic subvariety of S S SSS, defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯.
It is interesting to notice that Conjecture 5.11, which is stronger than Theorem 5.12, predicts the existence of a Hodge generic Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯-point in S S SSS for V V V\mathbb{V}V in the situation of Corollary 6.3.
As the criterion given in [57] is purely geometric, it says nothing about fields of definitions of special points. It is, however, strong enough to reduce the first part of Conjecture 6.1 to this particular case:
Theorem 6.4. Special subvarieties for Z V H S Z V H S ZVHS\mathbb{Z} V H SZVHS defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯ are defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯ if and only if it holds true for special points.

6.2. Absolute Hodge locus

Interestingly, Conjecture 6.1 in the geometric case follows from an a priori much weaker conjecture than the Hodge conjecture. Let f : X S f : X → S f:X rarr Sf: X \rightarrow Sf:X→S be a smooth projective morphism of smooth connected complex quasiprojective varieties. For any automorphism σ σ ∈ sigma in\sigma \inσ∈ Aut ( C / Q ) Aut ⁡ ( C / Q ) Aut(C//Q)\operatorname{Aut}(\mathbb{C} / \mathbb{Q})Aut⁡(C/Q), we can consider the algebraic family f σ : X σ S σ f σ : X σ → S σ f^(sigma):X^(sigma)rarrS^(sigma)f^{\sigma}: X^{\sigma} \rightarrow S^{\sigma}fσ:Xσ→Sσ, where σ 1 : S σ = S × C , σ σ − 1 : S σ = S × C , σ sigma^(-1):S^(sigma)=Sxx_(C,sigma)\sigma^{-1}: S^{\sigma}=S \times_{\mathbb{C}, \sigma}σ−1:Sσ=S×C,σ C S C → ∼ S Crarr"∼"S\mathbb{C} \xrightarrow{\sim} SC→∼S is the natural isomorphism of abstract schemes; and the attached polarizable Z V S H Z V S H ZVSH\mathbb{Z V S H}ZVSH V σ = ( V Z σ , V σ , F σ , σ ) V σ = V Z σ , V σ , F ∙ σ , ∇ σ V^(sigma)=(V_(Z)^(sigma),V^(sigma),F^(∙sigma),grad^(sigma))\mathbb{V}^{\sigma}=\left(\mathbb{V}_{\mathbb{Z}}^{\sigma}, \mathcal{V}^{\sigma}, F^{\bullet \sigma}, \nabla^{\sigma}\right)Vσ=(VZσ,Vσ,F∙σ,∇σ) with underlying local system V Z σ = R f σ an Z V Z σ = R f σ ∗ an  Z V_(Z)^(sigma)=Rf^(sigma)_(**)^("an ")Z\mathbb{V}_{\mathbb{Z}}^{\sigma}=R f^{\sigma}{ }_{*}^{\text {an }} \mathbb{Z}VZσ=Rfσ∗an Z on ( S σ ) an S σ an  (S^(sigma))^("an ")\left(S^{\sigma}\right)^{\text {an }}(Sσ)an . The algebraic construction of the algebraic de Rham cohomology provides compatible canonical comparison isomorphisms ι σ : ( V σ , F σ , σ ) σ 1 ( V , F , ) ι σ : V σ , F ∙ σ , ∇ σ → ∼ σ − 1 ∗ V , F ∙ , ∇ iota^(sigma):(V^(sigma),F^(∙sigma),grad^(sigma))rarr"∼"sigma^(-1^(**))(V,F^(∙),grad)\iota^{\sigma}:\left(\mathcal{V}^{\sigma}, F^{\bullet \sigma}, \nabla^{\sigma}\right) \xrightarrow{\sim} \sigma^{-1^{*}}\left(\mathcal{V}, F^{\bullet}, \nabla\right)ισ:(Vσ,F∙σ,∇σ)→∼σ−1∗(V,F∙,∇) of the associated algebraic filtered vector bundles with connection. More generally, a collection of Z V H S ( V σ ) σ Z V H S V σ σ ZVHS(V^(sigma))_(sigma)\mathbb{Z V H S}\left(\mathbb{V}^{\sigma}\right)_{\sigma}ZVHS(Vσ)σ with such compatible comparison isomorphisms is called a (de Rham) motivic variation of Hodge structures on S S SSS, in which case we write V := V Id V := V Id  V:=V^("Id ")\mathbb{V}:=\mathbb{V}^{\text {Id }}V:=VId . Following Deligne (see [25] for a nice exposition), an absolute Hodge tensor for such a collection is a Hodge tensor α α alpha\alphaα for V s V s V_(s)\mathbb{V}_{s}Vs such that the conjugates σ 1 α d R σ − 1 ∗ α d R sigma^(-1^(**))alpha_(dR)\sigma^{-1^{*}} \alpha_{\mathrm{dR}}σ−1∗αdR of the de Rham component of α α alpha\alphaα defines a Hodge tensor in V σ ( s ) σ V σ ( s ) σ V_(sigma(s))^(sigma)\mathbb{V}_{\sigma(s)}^{\sigma}Vσ(s)σ for all σ σ sigma\sigmaσ. The generic absolute Mumford-Tate group for ( V σ ) σ V σ σ (V^(sigma))_(sigma)\left(\mathbb{V}^{\sigma}\right)_{\sigma}(Vσ)σ is defined in terms of the absolute Hodge tensors as the generic Mumford-Tate group is defined in terms of the Hodge tensors. Thus G G A H G ⊂ G A H GsubG^(AH)\mathbf{G} \subset \mathbf{G}^{\mathrm{AH}}G⊂GAH. In view of Definition 3.19 the following is natural:
Definition 6.5. Let ( V σ ) σ V σ σ (V^(sigma))_(sigma)\left(\mathbb{V}^{\sigma}\right)_{\sigma}(Vσ)σ be a (de Rham) motivic variation of Hodge structure on a smooth connected complex quasiprojective variety S S SSS. A closed irreducible algebraic subvariety Y Y YYY of S S SSS is called absolutely special if it is maximal among the closed irreducible algebraic subvarieties Z Z ZZZ of S S SSS satisfying G Z A H = G Y A H G Z A H = G Y A H G_(Z)^(AH)=G_(Y)^(AH)\mathbf{G}_{Z}^{\mathrm{AH}}=\mathbf{G}_{Y}^{\mathrm{AH}}GZAH=GYAH.
In the geometric case, the Hodge conjecture implies, since any automorphism σ σ ∈ sigma in\sigma \inσ∈ Aut ( C / Q ) Aut ⁡ ( C / Q ) Aut(C//Q)\operatorname{Aut}(\mathbb{C} / \mathbb{Q})Aut⁡(C/Q) maps algebraic cycles in X X XXX to algebraic cycles on X σ X σ X^(sigma)X^{\sigma}Xσ, the following conjecture of Deligne:
Conjecture 6.6 ([33]). Let ( V σ ) σ V σ σ (V^(sigma))_(sigma)\left(\mathbb{V}^{\sigma}\right)_{\sigma}(Vσ)σ be a (de Rham) motivic variation of Hodge structure on S. Then all Hodge tensors are absolute Hodge tensors, i.e., G = G A H G = G A H G=G^(AH)\mathbf{G}=\mathbf{G}^{\mathrm{AH}}G=GAH.
This conjecture immediately implies:
Conjecture 6.7. Let ( V σ ) σ V σ σ (V^(sigma))_(sigma)\left(\mathbb{V}^{\sigma}\right)_{\sigma}(Vσ)σ be a (de Rham) motivic variation of Hodge structure on S S SSS. Then any special subvariety of S S SSS for V V V\mathbb{V}V is absolutely special for ( V σ ) σ V σ σ (V^(sigma))_(sigma)\left(\mathbb{V}^{\sigma}\right)_{\sigma}(Vσ)σ.
Let us say that a (de Rham) motivic variation ( V σ ) σ V σ σ (V^(sigma))_(sigma)\left(\mathbb{V}^{\sigma}\right)_{\sigma}(Vσ)σ is defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯ if V σ = V σ = V^(sigma)=\mathbb{V}^{\sigma}=Vσ= V V V\mathbb{V}V for all σ Aut ( C / Q ¯ ) σ ∈ Aut ⁡ ( C / Q ¯ ) sigma in Aut(C// bar(Q))\sigma \in \operatorname{Aut}(\mathbb{C} / \overline{\mathbb{Q}})σ∈Aut⁡(C/Q¯). In the geometric case, any morphism f : X S f : X → S f:X rarr Sf: X \rightarrow Sf:X→S defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯ defines such a (de Rham) motivic variation ( V σ ) σ V σ σ (V^(sigma))_(sigma)\left(\mathbb{V}^{\sigma}\right)_{\sigma}(Vσ)σ over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯. Notice that the absolutely
special subvarieties of S S SSS for ( V σ ) σ V σ σ (V^(sigma))_(sigma)\left(\mathbb{V}^{\sigma}\right)_{\sigma}(Vσ)σ are then by their very definition defined over Q ¯ Q ¯ bar(Q)\overline{\mathbb{Q}}Q¯, and their Galois conjugates are also special. In particular, Conjecture 6.7 implies Conjecture 6.1 in the geometric case. As proven in [95], Deligne's conjecture is actually equivalent to a much stronger version of Conjecture 6.1, where one replaces the special subvarieties of S S SSS (components of the Hodge locus) with the special subvarieties in the total bundle of V V ⊗ V^(ox)\mathcal{V}^{\otimes}V⊗ (components of the locus of Hodge tensors).
Recently T. Kreutz, using the same geometric argument as in [57], justified Theorem 6.2 by proving:
Theorem 6.8 ([62]). Let ( V σ ) σ V σ σ (V^(sigma))_(sigma)\left(\mathbb{V}^{\sigma}\right)_{\sigma}(Vσ)σ be a (de Rham) motivic variation of Hodge structure on S S SSS. Suppose that the adjoint generic Mumford-Tate group G ad G ad  G^("ad ")\mathbf{G}^{\text {ad }}Gad  is simple. Then any strict maximal special subvariety Y S Y ⊂ S Y sub SY \subset SY⊂S of positive period dimension for V V V\mathbb{V}V is absolutely special.
We refer the reader to [61], as well as [93], for other arithmetic aspects of Hodge loci taking into account not only the de Rham incarnation of absolute Hodge classes but also their â„“ â„“\ellâ„“-adic components.

ACKNOWLEDGMENTS

I would like to thank Gregorio Baldi, Benjamin Bakker, Yohan Brunebarbe, Jeremy Daniel, Philippe Eyssidieux, Ania Otwinowska, Carlos Simpson, Emmanuel Ullmo, Claire Voisin, and Andrei Yafaev for many interesting discussions on Hodge theory. I also thank Gregorio Baldi, Tobias Kreutz, and Leonardo Lerer for their comments on this text.

FUNDING

This work was partially supported by the ERC Advanced Grant 101020009 "TameHodge."

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BRUNO KLINGLER

Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany, bruno.klingler@hu-berlin.de

CANONICAL KÄHLER METRICS AND STABILITY OF ALGEBRAIC VARIETIES

CHI LI

ABSTRACT

We survey some recent developments in the study of canonical Kähler metrics on algebraic varieties and their relation with stability in algebraic geometry.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 53C55; Secondary 32Q26, 14J45, 32Q20, 58E11, 32P05

KEYWORDS

Constant scalar curvature Kähler metrics, Kähler-Einstein metrics, K-stability, valuative criterion, non-Archimedean geometry, Yau-Tian-Donaldson conjecture
The study of canonical Kähler metrics on algebraic varieties is a very active program in complex geometry. It is a common playground of several fields: differential geometry, partial differential equations, pluripotential theory, birational algebraic geometry, and nonArchimedean analysis. We will try to give the reader a tour of this vast program, emphasizing recent developments and highlighting interactions of different concepts and techniques. This article consists of three parts. In the first part, we discuss important classes of canonical Kähler metrics, and explain a well-established variational formalism for studying their existence. In the second part, we discuss algebraic aspects by reviewing recent developments in the study of K-stability with the help of deep tools from algebraic geometry and nonArchimedean analysis. In the third part, we discuss how the previous two parts are connected with each other. In particular, we will discuss the Yau-Tian-Donaldson (YTD) conjecture for canonical Kähler metrics in the first part.

1. CANONICAL KÄHLER METRICS ON ALGEBRAIC VARIETIES

1.1. Constant scalar curvature Kähler metrics

Let X X XXX be an n n nnn-dimensional projective manifold equipped with an ample line bundle L L LLL. By Kodaira's theorem, we have an embedding ι m : X P N ι m : X → P N iota_(m):X rarrP^(N)\iota_{m}: X \rightarrow \mathbb{P}^{N}ιm:X→PN by using a complete linear system | m L | | m L | |mL||m L||mL| for m 1 m ≫ 1 m≫1m \gg 1m≫1. If we denote by h F S h F S h_(FS)h_{\mathrm{FS}}hFS the standard Fubini-Study metric on the hyperplane bundle over P N P N P^(N)\mathbb{P}^{N}PN with Chern curvature ω F S = d d c log h F S ω F S = − d d c log ⁡ h F S omega_(FS)=-dd^(c)log h_(FS)\omega_{\mathrm{FS}}=-\mathrm{dd}^{\mathrm{c}} \log h_{\mathrm{FS}}ωFS=−ddclog⁡hFS, then h 0 = ι m h F S 1 / m h 0 = ι m ∗ h F S 1 / m h_(0)=iota_(m)^(**)h_(FS)^(1//m)h_{0}=\iota_{m}^{*} h_{\mathrm{FS}}^{1 / m}h0=ιm∗hFS1/m is a smooth Hermitian metric on L L LLL whose Chern curvature ω 0 = 1 m ι m ω F S = d d c log h 0 ω 0 = 1 m ι m ∗ ω F S = − d d c log ⁡ h 0 omega_(0)=(1)/(m)iota_(m)^(**)omega_(FS)=-dd^(c)log h_(0)\omega_{0}=\frac{1}{m} \iota_{m}^{*} \omega_{\mathrm{FS}}=-\mathrm{dd}^{\mathrm{c}} \log h_{0}ω0=1mιm∗ωFS=−ddclog⁡h0 is a Kähler form in c 1 ( L ) H 2 ( X , R ) c 1 ( L ) ∈ H 2 ( X , R ) c_(1)(L)inH^(2)(X,R)c_{1}(L) \in H^{2}(X, \mathbb{R})c1(L)∈H2(X,R). In this paper we will use the convention d d c = 1 2 π ¯ d d c = − 1 2 Ï€ ∂ ∂ ¯ dd^(c)=(sqrt(-1))/(2pi)del bar(del)\mathrm{dd}^{\mathrm{c}}=\frac{\sqrt{-1}}{2 \pi} \partial \bar{\partial}ddc=−12π∂∂¯.
We will also use singular Hermitian metrics. An upper-semicontinuous function φ L 1 ( ω n ) φ ∈ L 1 ω n varphi inL^(1)(omega^(n))\varphi \in L^{1}\left(\omega^{n}\right)φ∈L1(ωn) is called an ω 0 ω 0 omega_(0)\omega_{0}ω0-psh potential if ψ + φ ψ + φ psi+varphi\psi+\varphiψ+φ is a plurisubharmonic function for any local potential ψ ψ psi\psiψ of ω 0 ω 0 omega_(0)\omega_{0}ω0 (i.e., ω 0 = dd c ψ ω 0 = dd c ⁡ ψ omega_(0)=dd^(c)psi\omega_{0}=\operatorname{dd}^{\mathrm{c}} \psiω0=ddc⁡ψ locally); h φ := h 0 e φ h φ := h 0 e − φ h_(varphi):=h_(0)e^(-varphi)h_{\varphi}:=h_{0} e^{-\varphi}hφ:=h0e−φ is then called a psh Hermitian metric on L L LLL. Denote by PSH ( ω 0 ) PSH ⁡ ω 0 PSH(omega_(0))\operatorname{PSH}\left(\omega_{0}\right)PSH⁡(ω0) the space of ω 0 ω 0 omega_(0)\omega_{0}ω0-psh functions. By a ¯ ∂ ∂ ¯ del bar(del)\partial \bar{\partial}∂∂¯-lemma, any closed positive (1,1)-current in c 1 ( L ) c 1 ( L ) c_(1)(L)c_{1}(L)c1(L) is of the form ω φ := ω 0 + d d c φ = d d c log h φ ω φ := ω 0 + d d c φ = − d d c log ⁡ h φ omega_(varphi):=omega_(0)+dd^(c)varphi=-dd^(c)log h_(varphi)\omega_{\varphi}:=\omega_{0}+\mathrm{dd}^{\mathrm{c}} \varphi=-\mathrm{dd}^{\mathrm{c}} \log h_{\varphi}ωφ:=ω0+ddcφ=−ddclog⁡hφ with φ φ ∈ varphi in\varphi \inφ∈ PSH ( ω 0 ) PSH ⁡ ω 0 PSH(omega_(0))\operatorname{PSH}\left(\omega_{0}\right)PSH⁡(ω0). Moreover, ω φ 2 = ω φ 1 ω φ 2 = ω φ 1 omega_(varphi_(2))=omega_(varphi_(1))\omega_{\varphi_{2}}=\omega_{\varphi_{1}}ωφ2=ωφ1 if and only if φ 2 φ 1 φ 2 − φ 1 varphi_(2)-varphi_(1)\varphi_{2}-\varphi_{1}φ2−φ1 is a constant. Define the space of smooth strictly ω 0 ω 0 omega_(0)\omega_{0}ω0-psh potentials (also called Kähler potentials) by
(1.1) H := H ( ω 0 ) = { φ C ( X ) : ω φ = ω 0 + dd c φ > 0 } (1.1) H := H ω 0 = φ ∈ C ∞ ( X ) : ω φ = ω 0 + dd c ⁡ φ > 0 {:(1.1)H:=H(omega_(0))={varphi inC^(oo)(X):omega_(varphi)=omega_(0)+dd^(c)varphi > 0}:}\begin{equation*} \mathscr{H}:=\mathscr{H}\left(\omega_{0}\right)=\left\{\varphi \in C^{\infty}(X): \omega_{\varphi}=\omega_{0}+\operatorname{dd}^{\mathrm{c}} \varphi>0\right\} \tag{1.1} \end{equation*}(1.1)H:=H(ω0)={φ∈C∞(X):ωφ=ω0+ddc⁡φ>0}
Fix any φ H φ ∈ H varphi inH\varphi \in \mathscr{H}φ∈H. If ω φ = 1 i , j ( ω φ ) i j d z i d z ¯ j ω φ = − 1 ∑ i , j   ω φ i j d z i ∧ d z ¯ j omega_(varphi)=sqrt(-1)sum_(i,j)(omega_(varphi))_(ij)dz_(i)^^d bar(z)_(j)\omega_{\varphi}=\sqrt{-1} \sum_{i, j}\left(\omega_{\varphi}\right)_{i j} d z_{i} \wedge d \bar{z}_{j}ωφ=−1∑i,j(ωφ)ijdzi∧dz¯j under a holomorphic coordinate chart, then its Ricci curvature form Ric ( ω φ ) = 1 2 π i , j R i j ¯ d z i d z ¯ j Ric ⁡ ω φ = − 1 2 Ï€ ∑ i , j   R i j ¯ d z i ∧ d z ¯ j Ric(omega_(varphi))=(sqrt(-1))/(2pi)sum_(i,j)R_(i bar(j))dz_(i)^^d bar(z)_(j)\operatorname{Ric}\left(\omega_{\varphi}\right)=\frac{\sqrt{-1}}{2 \pi} \sum_{i, j} R_{i \bar{j}} d z_{i} \wedge d \bar{z}_{j}Ric⁡(ωφ)=−12π∑i,jRij¯dzi∧dz¯j is given by
R i j ¯ := Ric ( ω φ ) i j ¯ = 2 log det ( ( ω φ ) k l ¯ ) z i z ¯ j R i j ¯ := Ric ⁡ ω φ i j ¯ = − ∂ 2 log ⁡ det ⁡ ω φ k l ¯ ∂ z i ∂ z ¯ j R_(i bar(j)):=Ric (omega_(varphi))_(i bar(j))=-(del^(2)log det((omega_(varphi))_(k bar(l))))/(delz_(i)del bar(z)_(j))R_{i \bar{j}}:=\operatorname{Ric}\left(\omega_{\varphi}\right)_{i \bar{j}}=-\frac{\partial^{2} \log \operatorname{det}\left(\left(\omega_{\varphi}\right)_{k \bar{l}}\right)}{\partial z_{i} \partial \bar{z}_{j}}Rij¯:=Ric⁡(ωφ)ij¯=−∂2log⁡det⁡((ωφ)kl¯)∂zi∂z¯j
Then Ric ( ω φ ) Ric ⁡ ω φ Ric(omega_(varphi))\operatorname{Ric}\left(\omega_{\varphi}\right)Ric⁡(ωφ) is a real closed ( 1 , 1 ) ( 1 , 1 ) (1,1)(1,1)(1,1)-form which represents the cohomology class c 1 ( K X ) = c 1 − K X = c_(1)(-K_(X))=c_{1}\left(-K_{X}\right)=c1(−KX)= : c 1 ( X ) c 1 ( X ) c_(1)(X)c_{1}(X)c1(X). Here K X = n T ( 1 , 0 ) X − K X = ∧ n T ( 1 , 0 ) X -K_(X)=^^^(n)T^((1,0))X-K_{X}=\wedge^{n} T^{(1,0)} X−KX=∧nT(1,0)X is the anticanonical line bundle of X X XXX. The scalar curvature of ω φ ω φ omega_(varphi)\omega_{\varphi}ωφ is given by the contraction
S ( ω φ ) = ω φ i j ¯ ( Ric ( ω φ ) ) i j ¯ = n Ric ( ω φ ) ω φ n 1 ω φ n S ω φ = ω φ i j ¯ Ric ⁡ ω φ i j ¯ = n â‹… Ric ⁡ ω φ ∧ ω φ n − 1 ω φ n S(omega_(varphi))=omega_(varphi)^(i bar(j))(Ric(omega_(varphi)))_(i bar(j))=(n*Ric(omega_(varphi))^^omega_(varphi)^(n-1))/(omega_(varphi)^(n))S\left(\omega_{\varphi}\right)=\omega_{\varphi}^{i \bar{j}}\left(\operatorname{Ric}\left(\omega_{\varphi}\right)\right)_{i \bar{j}}=\frac{n \cdot \operatorname{Ric}\left(\omega_{\varphi}\right) \wedge \omega_{\varphi}^{n-1}}{\omega_{\varphi}^{n}}S(ωφ)=ωφij¯(Ric⁡(ωφ))ij¯=nâ‹…Ric⁡(ωφ)∧ωφn−1ωφn
Further, ω φ ω φ omega_(varphi)\omega_{\varphi}ωφ is called a constant scalar curvature Kähler ( cscK ) ( cscK ) (cscK)(\operatorname{cscK})(cscK) metric if S ( ω φ ) S ω φ S(omega_(varphi))S\left(\omega_{\varphi}\right)S(ωφ) is the constant S _ S _ S_\underline{S}S_ which is the average scalar curvature and is determined by cohomology classes:
(1.2) S _ = n c 1 ( X ) c 1 ( L ) n 1 , [ X ] V with V = c 1 ( L ) n , [ X ] (1.2) S _ = n c 1 ( X ) ⋅ c 1 ( L ) ⋅ n − 1 , [ X ] V  with  V = c 1 ( L ) ⋅ n , [ X ] {:(1.2)S_=(n(:c_(1)(X)*c_(1)(L)^(*n-1),[X]:))/(V)quad" with "V=(:c_(1)(L)^(*n),[X]:):}\begin{equation*} \underline{S}=\frac{n\left\langle c_{1}(X) \cdot c_{1}(L)^{\cdot n-1},[X]\right\rangle}{\mathbf{V}} \quad \text { with } \mathbf{V}=\left\langle c_{1}(L)^{\cdot n},[X]\right\rangle \tag{1.2} \end{equation*}(1.2)S_=n⟨c1(X)⋅c1(L)⋅n−1,[X]⟩V with V=⟨c1(L)⋅n,[X]⟩
The Kähler potential of a cscK metric is a solution to a 4th order nonlinear PDE. In general, there are obstructions to the existence of cscK metrics. For example, the MatsushimaLichnerowicz theorem states that if ( X , L ) ( X , L ) (X,L)(X, L)(X,L) admits a cscK metric then the automorphism group Aut ( X , L ) Aut ⁡ ( X , L ) Aut(X,L)\operatorname{Aut}(X, L)Aut⁡(X,L) must be reductive. Our goal is to discuss the Yau-Tian-Donaldson conjecture which would provide a sufficient and necessary algebraic criterion for the existence of cscK cscK cscK\operatorname{cscK}cscK metrics.

1.2. Kähler-Einstein metrics and weighted Kähler-Ricci soliton

Kähler-Einstein metrics form an important class of cscK metrics. A Kähler form ω φ ω φ omega_(varphi)\omega_{\varphi}ωφ is called Kähler-Einstein ( K E ) ( K E ) (KE)(\mathrm{KE})(KE) if Ric ( ω φ ) = λ ω φ Ric ⁡ ω φ = λ ω φ Ric(omega_(varphi))=lambdaomega_(varphi)\operatorname{Ric}\left(\omega_{\varphi}\right)=\lambda \omega_{\varphi}Ric⁡(ωφ)=λωφ for a real constant λ λ lambda\lambdaλ. A necessary condition for the existence of K E K E KE\mathrm{KE}KE metrics is that the cohomology class c 1 ( X ) H 2 ( X , R ) c 1 ( X ) ∈ H 2 ( X , R ) c_(1)(X)inH^(2)(X,R)c_{1}(X) \in H^{2}(X, \mathbb{R})c1(X)∈H2(X,R) is either negative, numerically trivial, or positive. The existence for the first two cases was understood in 1970s: there always exists a Kähler-Einstein metric if c 1 ( X ) c 1 ( X ) c_(1)(X)c_{1}(X)c1(X) is negative (by the work of Aubin and Yau), or if c 1 ( X ) c 1 ( X ) c_(1)(X)c_{1}(X)c1(X) is numerically trivial (by the work of Yau).
Now we assume that X X XXX is a Fano manifold. In other words, K X − K X -K_(X)-K_{X}−KX is an ample line bundle, and we set L = K X L = − K X L=-K_(X)L=-K_{X}L=−KX. Any φ H φ ∈ H varphi inH\varphi \in \mathscr{H}φ∈H corresponds to a volume form
Ω φ := | s | h φ 2 ( 1 ) n 2 s s ¯ = Ω 0 e φ with s = d z 1 d z n , s = z 1 z n Ω φ := s ∗ h φ 2 ( − 1 ) n 2 s ∧ s ¯ = Ω 0 e − φ  with  s = d z 1 ∧ ⋯ ∧ d z n , s ∗ = ∂ z 1 ∧ ⋯ ∧ ∂ z n Omega_(varphi):=|s^(**)|_(h_(varphi))^(2)(sqrt(-1))^(n^(2))s^^ bar(s)=Omega_(0)e^(-varphi)quad" with "s=dz_(1)^^cdots^^dz_(n),s^(**)=del_(z_(1))^^cdots^^del_(z_(n))\Omega_{\varphi}:=\left|s^{*}\right|_{h_{\varphi}}^{2}(\sqrt{-1})^{n^{2}} s \wedge \bar{s}=\Omega_{0} e^{-\varphi} \quad \text { with } s=d z_{1} \wedge \cdots \wedge d z_{n}, s^{*}=\partial_{z_{1}} \wedge \cdots \wedge \partial_{z_{n}}Ωφ:=|s∗|hφ2(−1)n2s∧s¯=Ω0e−φ with s=dz1∧⋯∧dzn,s∗=∂z1∧⋯∧∂zn
The K E K E KE\mathrm{KE}KE equation in this case is reduced to a complex Monge-Ampère equation for φ φ varphi\varphiφ, namely
( ω + dd c φ ) n = e φ Ω 0 ω + dd c ⁡ φ n = e − φ Ω 0 (omega+dd^(c)varphi)^(n)=e^(-varphi)Omega_(0)\left(\omega+\operatorname{dd}^{\mathrm{c}} \varphi\right)^{n}=e^{-\varphi} \Omega_{0}(ω+ddc⁡φ)n=e−φΩ0
We also consider an interesting generalization of Kähler-Einstein metrics on Fano manifolds with torus actions. Assume that T ( C ) r T ≅ C ∗ r T~=(C^(**))^(r)\mathbb{T} \cong\left(\mathbb{C}^{*}\right)^{r}T≅(C∗)r is an algebraic torus and T ( S 1 ) r T T ≅ S 1 r ⊂ T T~=(S^(1))^(r)subTT \cong\left(S^{1}\right)^{r} \subset \mathbb{T}T≅(S1)r⊂T is a compact real subtorus. We will use the following notation:
(1.3) N Z = Hom a l g ( C , T ) , N Q = N Z Z Q , N R = N Z Z R (1.3) N Z = Hom a l g ⁡ C ∗ , T , N Q = N Z ⊗ Z Q , N R = N Z ⊗ Z R {:(1.3)N_(Z)=Hom_(alg)(C^(**),T)","quadN_(Q)=N_(Z)ox_(Z)Q","quadN_(R)=N_(Z)ox_(Z)R:}\begin{equation*} N_{\mathbb{Z}}=\operatorname{Hom}_{\mathrm{alg}}\left(\mathbb{C}^{*}, \mathbb{T}\right), \quad N_{\mathbb{Q}}=N_{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Q}, \quad N_{\mathbb{R}}=N_{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{R} \tag{1.3} \end{equation*}(1.3)NZ=Homalg⁡(C∗,T),NQ=NZ⊗ZQ,NR=NZ⊗ZR
Assume that T T T\mathbb{T}T acts faithfully on X X XXX. Then there is an induced T T T\mathbb{T}T-action on K X − K X -K_(X)-K_{X}−KX. Each ξ N R ξ ∈ N R xi inN_(R)\xi \in N_{\mathbb{R}}ξ∈NR corresponds to a holomorphic vector field V ξ V ξ V_(xi)V_{\xi}Vξ on X X XXX. Denote by H T H T H^(T)\mathscr{H}^{T}HT the set of T T TTT-invariant Kähler potentials. For any φ H T φ ∈ H T varphi inH^(T)\varphi \in \mathscr{H}^{T}φ∈HT, the T T TTT-action becomes Hamiltonian with respect to ω φ ω φ omega_(varphi)\omega_{\varphi}ωφ. Denote by m φ : X N R R r m φ : X → N R ∗ ≅ R r m_(varphi):X rarrN_(R)^(**)~=R^(r)\mathbf{m}_{\varphi}: X \rightarrow N_{\mathbb{R}}^{*} \cong \mathbb{R}^{r}mφ:X→NR∗≅Rr the corresponding moment map, and let P P PPP be the image of m φ m φ m_(varphi)\mathbf{m}_{\varphi}mφ. By a theorem of Atiyah-Guillemin-Sternberg, P P PPP is a convex polytope which depends only on the Kähler class c 1 ( L ) c 1 ( L ) c_(1)(L)c_{1}(L)c1(L). Let g : P R g : P → R g:P rarrRg: P \rightarrow \mathbb{R}g:P→R be a smooth positive function. The following equation will be called the g g ggg-weighted soliton (or just g g ggg-soliton) equation for φ H ( K X ) T φ ∈ H − K X T varphi inH(-K_(X))^(T)\varphi \in \mathscr{H}\left(-K_{X}\right)^{T}φ∈H(−KX)T.
g ( m φ ) ( ω 0 + dd c φ ) n = e φ Ω 0 g m φ ω 0 + dd c ⁡ φ n = e − φ Ω 0 g(m_(varphi))(omega_(0)+dd^(c)varphi)^(n)=e^(-varphi)Omega_(0)g\left(\mathbf{m}_{\varphi}\right)\left(\omega_{0}+\operatorname{dd}^{\mathrm{c}} \varphi\right)^{n}=e^{-\varphi} \Omega_{0}g(mφ)(ω0+ddc⁡φ)n=e−φΩ0
An equivalent tensorial equation is given by Ric ( ω φ ) = ω φ + d d c log g ( m φ ) Ric ⁡ ω φ = ω φ + d d c log ⁡ g m φ Ric(omega_(varphi))=omega_(varphi)+dd^(c)log g(m_(varphi))\operatorname{Ric}\left(\omega_{\varphi}\right)=\omega_{\varphi}+\mathrm{dd}^{\mathrm{c}} \log g\left(\mathbf{m}_{\varphi}\right)Ric⁡(ωφ)=ωφ+ddclog⁡g(mφ).
Example 1.1. If g ( y ) = e y , ξ g ( y ) = e − ⟨ y , ξ ⟩ g(y)=e^(-(:y,xi:))g(y)=e^{-\langle y, \xi\rangle}g(y)=e−⟨y,ξ⟩, then the above equation becomes the standard Kähler-Ricci soliton equation Ric ( ω φ ) = ω φ + L V ξ ω φ Ric ⁡ ω φ = ω φ + L V ξ ω φ Ric(omega_(varphi))=omega_(varphi)+L_(V_(xi))omega_(varphi)\operatorname{Ric}\left(\omega_{\varphi}\right)=\omega_{\varphi}+\mathscr{L}_{V_{\xi}} \omega_{\varphi}Ric⁡(ωφ)=ωφ+LVξωφ where L L L\mathscr{L}L denotes the Lie derivative.

1.3. Kähler-Einstein metrics on log Fano pairs

Singular algebraic varieties and log pairs are important objects in algebraic geometry, and appear naturally for studying limits of smooth varieties. It is thus natural to study canonical Kähler metric on general log pairs. We recall a definition from birational algebraic geometry. Let X X XXX be a normal projective variety and D D DDD be a Q Q Q\mathbb{Q}Q-Weil divisor. Assume that K X + D K X + D K_(X)+DK_{X}+DKX+D is Q Q Q\mathbb{Q}Q-Cartier. Let μ : Y X μ : Y → X mu:Y rarr X\mu: Y \rightarrow Xμ:Y→X be a resolution of singularities of ( X , D ) ( X , D ) (X,D)(X, D)(X,D) with simple normal crossing exceptional divisors i E i ∑ i   E i sum_(i)E_(i)\sum_{i} E_{i}∑iEi. We then have an identity
(1.4) K Y = μ ( K X + D ) + i a i E i (1.4) K Y = μ ∗ K X + D + ∑ i   a i E i {:(1.4)K_(Y)=mu^(**)(K_(X)+D)+sum_(i)a_(i)E_(i):}\begin{equation*} K_{Y}=\mu^{*}\left(K_{X}+D\right)+\sum_{i} a_{i} E_{i} \tag{1.4} \end{equation*}(1.4)KY=μ∗(KX+D)+∑iaiEi
Here A ( X , D ) ( E i ) := a i + 1 A ( X , D ) E i := a i + 1 A_((X,D))(E_(i)):=a_(i)+1A_{(X, D)}\left(E_{i}\right):=a_{i}+1A(X,D)(Ei):=ai+1 is called the log discrepancy of E i E i E_(i)E_{i}Ei. The pair ( X , D ) ( X , D ) (X,D)(X, D)(X,D) has klt singularities if A ( X , D ) ( E i ) > 0 A ( X , D ) E i > 0 A_((X,D))(E_(i)) > 0A_{(X, D)}\left(E_{i}\right)>0A(X,D)(Ei)>0 for any E i E i E_(i)E_{i}Ei. We will always assume that ( X , D ) ( X , D ) (X,D)(X, D)(X,D) has klt singularities.
If K X + D K X + D K_(X)+DK_{X}+DKX+D is ample or numerically trivial, Yau and Aubin's existence result had been generalized to the singular and log case in [32], partly based on Kołodziej's pluripotential estimates. There were many related works by Yau, Tian, H. Tsuji, Z. Zhang, and many others.
Now we assume that ( K X + D ) − K X + D -(K_(X)+D)-\left(K_{X}+D\right)−(KX+D) is ample and call ( X , D ) ( X , D ) (X,D)(X, D)(X,D) a log Fano pair. Then one can consider Kähler-Einstein equation or, more generally, g g ggg-soliton equation on ( X , D ) ( X , D ) (X,D)(X, D)(X,D). Note that there is a globally defined volume form as in the smooth case: choose a local trivializing section s s sss of m ( K X + D ) m K X + D m(K_(X)+D)m\left(K_{X}+D\right)m(KX+D) with the dual s s ∗ s^(**)s^{*}s∗ and define Ω 0 = | s | h 0 2 / m ( 1 m n 2 s Ω 0 = s ∗ h 0 2 / m − 1 m n 2 s ∧ Omega_(0)=|s^(**)|_(h_(0))^(2//m)(sqrt-1^(mn^(2))s^^:}\Omega_{0}=\left|s^{*}\right|_{h_{0}}^{2 / m}\left(\sqrt{-1}^{m n^{2}} s \wedge\right.Ω0=|s∗|h02/m(−1mn2s∧ s ¯ ) 1 / m s ¯ ) 1 / m bar(s))^(1//m)\bar{s})^{1 / m}s¯)1/m. Assume that T T T\mathbb{T}T acts on X X XXX faithfully and preserves the divisor D D DDD. With the notation from before, we say that φ φ varphi\varphiφ is the potential for a g g ggg-weighted soliton on ( X , D ) ( X , D ) (X,D)(X, D)(X,D) if φ φ varphi\varphiφ is a bounded ω 0 ω 0 omega_(0)\omega_{0}ω0-psh function that satisfies the equation
(1.5) g ( m φ ) ( ω + dd c φ ) n = e φ Ω 0 (1.5) g m φ ω + dd c ⁡ φ n = e − φ Ω 0 {:(1.5)g(m_(varphi))(omega+dd^(c)varphi)^(n)=e^(-varphi)Omega_(0):}\begin{equation*} g\left(\mathbf{m}_{\varphi}\right)\left(\omega+\operatorname{dd}^{\mathrm{c}} \varphi\right)^{n}=e^{-\varphi} \Omega_{0} \tag{1.5} \end{equation*}(1.5)g(mφ)(ω+ddc⁡φ)n=e−φΩ0
For any bounded φ PSH ( ω 0 ) φ ∈ PSH ⁡ ω 0 varphi in PSH(omega_(0))\varphi \in \operatorname{PSH}\left(\omega_{0}\right)φ∈PSH⁡(ω0), the g g ggg-weighted Monge-Ampère measure on the left-hand side of (1.5) is well defined by the work of Berman-Witt-Nyström [10] and also by HanLi [38], generalizing the definition of Bedford-Taylor (when g = 1 g = 1 g=1g=1g=1 ). It is known that any bounded solution φ φ varphi\varphiφ, if it exists, is orbifold smooth over the orbifold locus of ( X , D ) ( X , D ) (X,D)(X, D)(X,D). Moreover, if p p ppp is a regular point of supp ( D ) supp ⁡ ( D ) supp(D)\operatorname{supp}(D)supp⁡(D) such that D = ( 1 β ) { z 1 = 0 } D = ( 1 − β ) z 1 = 0 D=(1-beta){z_(1)=0}D=(1-\beta)\left\{z_{1}=0\right\}D=(1−β){z1=0} locally for a holomorphic function z 1 z 1 z_(1)z_{1}z1 (with β ( 0 , 1 ] β ∈ ( 0 , 1 ] beta in(0,1]\beta \in(0,1]β∈(0,1] ), then the associated Kähler metric is modeled by C β × C n 1 C β × C n − 1 C_(beta)xxC^(n-1)\mathbb{C}_{\beta} \times \mathbb{C}^{n-1}Cβ×Cn−1 where C β = ( C , d r 2 + β 2 r 2 d θ 2 ) C β = C , d r 2 + β 2 r 2 d θ 2 C_(beta)=(C,dr^(2)+beta^(2)r^(2)dtheta^(2))\mathbb{C}_{\beta}=\left(\mathbb{C}, d r^{2}+\beta^{2} r^{2} d \theta^{2}\right)Cβ=(C,dr2+β2r2dθ2) is the 2-dimensional flat cone with cone angle 2 π β 2 Ï€ β 2pi beta2 \pi \beta2πβ.

1.4. Ricci-flat Kähler cone metrics

The class of Ricci-flat Kähler cone metrics is closely related to K E / g K E / g KE//g\mathrm{KE} / \mathrm{g}KE/g-soliton metrics, and is interesting in both complex geometry and mathematical physics (see [57]).
Let Y = Spec ( R ) Y = Spec ⁡ ( R ) Y=Spec(R)Y=\operatorname{Spec}(R)Y=Spec⁡(R) be an ( n + 1 ) ( n + 1 ) (n+1)(n+1)(n+1)-dimensional normal affine variety with a singularity o Y o ∈ Y o in Yo \in Yo∈Y. Assume that an algebraic torus T ^ ( C ) r + 1 T ^ ≅ C ∗ r + 1 hat(T)~=(C^(**))^(r+1)\hat{\mathbb{T}} \cong\left(\mathbb{C}^{*}\right)^{r+1}T^≅(C∗)r+1 acts faithfully on Y Y YYY, with o o ooo being the only fixed point. Define N ^ Q , N ^ R N ^ Q , N ^ R hat(N)_(Q), hat(N)_(R)\hat{N}_{\mathbb{Q}}, \hat{N}_{\mathbb{R}}N^Q,N^R similar to (1.3). The T ^ T ^ hat(T)\hat{\mathbb{T}}T^-action corresponds to a weight decomposition of the coordinate ring R = α Z r + 1 R α R = ⨁ α ∈ Z r + 1   R α R=bigoplus_(alpha inZ^(r+1))R_(alpha)R=\bigoplus_{\alpha \in \mathbb{Z}^{r+1}} R_{\alpha}R=⨁α∈Zr+1Rα. The Reeb cone can be defined as
N ^ R + = { ξ N ^ R : α , ξ > 0 for all α Z r + 1 { 0 } with R α 0 } N ^ R + = ξ ∈ N ^ R : ⟨ α , ξ ⟩ > 0  for all  α ∈ Z r + 1 ∖ { 0 }  with  R α ≠ 0 hat(N)_(R)^(+)={xi in hat(N)_(R):(:alpha,xi:) > 0" for all "alpha inZ^(r+1)\\{0}" with "R_(alpha)!=0}\hat{N}_{\mathbb{R}}^{+}=\left\{\xi \in \hat{N}_{\mathbb{R}}:\langle\alpha, \xi\rangle>0 \text { for all } \alpha \in \mathbb{Z}^{r+1} \backslash\{0\} \text { with } R_{\alpha} \neq 0\right\}N^R+={ξ∈N^R:⟨α,ξ⟩>0 for all α∈Zr+1∖{0} with Rα≠0}
Any ξ ^ N ^ R + ξ ^ ∈ N ^ R + hat(xi)in hat(N)_(R)^(+)\hat{\xi} \in \hat{N}_{\mathbb{R}}^{+}ξ^∈N^R+is called a Reeb vector and corresponds to an expanding holomorphic vector field V ξ ^ V ξ ^ V_( hat(xi))V_{\hat{\xi}}Vξ^. Assume, furthermore, that Y Y YYY is Q Q Q\mathbb{Q}Q-Gorenstein and there is a T ^ T ^ hat(T)\hat{\mathbb{T}}T^-equivariant nonvanishing section s | m K Y | s ∈ m K Y s in|mK_(Y)|s \in\left|m K_{Y}\right|s∈|mKY|, which induces a T ^ T ^ hat(T)\hat{\mathbb{T}}T^-equivariant volume form d V Y = d V Y = dV_(Y)=d V_{Y}=dVY= ( 1 m ( n + 1 ) 2 s s ¯ ) 1 / m − 1 m ( n + 1 ) 2 s ∧ s ¯ 1 / m (sqrt-1^(m(n+1)^(2))s^^( bar(s)))^(1//m)\left(\sqrt{-1}^{m(n+1)^{2}} s \wedge \bar{s}\right)^{1 / m}(−1m(n+1)2s∧s¯)1/m on Y Y YYY. We call the data ( Y , ξ ^ ) ( Y , ξ ^ ) (Y, hat(xi))(Y, \hat{\xi})(Y,ξ^) with ξ ^ N ^ R + ξ ^ ∈ N ^ R + hat(xi)in hat(N)_(R)^(+)\hat{\xi} \in \hat{N}_{\mathbb{R}}^{+}ξ^∈N^R+a polarized Fano cone.
Let T ^ ( S 1 ) r + 1 T ^ ≅ S 1 r + 1 hat(T)~=(S^(1))^(r+1)\hat{T} \cong\left(S^{1}\right)^{r+1}T^≅(S1)r+1 be a compact real subtorus of T ^ T ^ hat(T)\hat{\mathbb{T}}T^. A T ^ T ^ hat(T)\hat{T}T^-invariant function r : Y r : Y → r:Y rarrr: Y \rightarrowr:Y→ R 0 R ≥ 0 R_( >= 0)\mathbb{R}_{\geq 0}R≥0 is called a radius function for ξ ^ N ^ R + ξ ^ ∈ N ^ R + hat(xi)in hat(N)_(R)^(+)\hat{\xi} \in \hat{N}_{\mathbb{R}}^{+}ξ^∈N^R+if ω ^ = d d c r 2 ω ^ = d d c r 2 widehat(omega)=dd^(c)r^(2)\widehat{\omega}=\mathrm{dd}^{\mathrm{c}} r^{2}ω^=ddcr2 is a Kähler cone metric on Y = Y ∗ = Y^(**)=Y^{*}=Y∗= Y { o } Y ∖ { o } Y\\{o}Y \backslash\{o\}Y∖{o} and 1 2 ( r r 1 J ( r r ) ) = V ξ ^ 1 2 r ∂ r − − 1 J r ∂ r = V ξ ^ (1)/(2)(rdel_(r)-sqrt(-1)J(rdel_(r)))=V_( hat(xi))\frac{1}{2}\left(r \partial_{r}-\sqrt{-1} J\left(r \partial_{r}\right)\right)=V_{\hat{\xi}}12(r∂r−−1J(r∂r))=Vξ^. Here J J JJJ is a complex structure on Y Y ∗ Y^(**)Y^{*}Y∗ and ω ^ ω ^ widehat(omega)\widehat{\omega}ω^ is called a Kähler cone metric if G := 1 2 ω ^ ( , J ) G := 1 2 ω ^ ( â‹… , J â‹… ) G:=(1)/(2) widehat(omega)(*,J*)G:=\frac{1}{2} \widehat{\omega}(\cdot, J \cdot)G:=12ω^(â‹…,Jâ‹…) on Y Y ∗ Y^(**)Y^{*}Y∗ is isometric to d r 2 + r 2 G S d r 2 + r 2 G S dr^(2)+r^(2)G_(S)d r^{2}+r^{2} G_{S}dr2+r2GS where S = { r = 1 } S = { r = 1 } S={r=1}S=\{r=1\}S={r=1} and G S = G | S G S = G S G_(S)=G|_(S)G_{S}=\left.G\right|_{S}GS=G|S. In the literature of CR geometry, the induced structure on the link S S SSS by a Kähler cone metric is called a Sasaki structure. Also ω ^ = dd c r 2 ω ^ = dd c ⁡ r 2 widehat(omega)=dd^(c)r^(2)\widehat{\omega}=\operatorname{dd}^{\mathrm{c}} r^{2}ω^=ddc⁡r2 is called Ricci-flat if Ric ( ω ^ ) = 0 Ric ⁡ ( ω ^ ) = 0 Ric( widehat(omega))=0\operatorname{Ric}(\widehat{\omega})=0Ric⁡(ω^)=0. In this case, the radius function satisfies the equation (up to rescaling)
( d d c r 2 ) n + 1 = d V Y d d c r 2 n + 1 = d V Y (dd^(c)r^(2))^(n+1)=dV_(Y)\left(\mathrm{dd}^{\mathrm{c}} r^{2}\right)^{n+1}=d V_{Y}(ddcr2)n+1=dVY
If ξ ^ N ^ Q ξ ^ ∈ N ^ Q hat(xi)in hat(N)_(Q)\hat{\xi} \in \hat{N}_{\mathbb{Q}}ξ^∈N^Q, then ω ^ ω ^ widehat(omega)\widehat{\omega}ω^ is called quasiregular, and V ξ ^ V ξ ^ V_( hat(xi))V_{\hat{\xi}}Vξ^ generates a C C ∗ C^(**)\mathbb{C}^{*}C∗-subgroup ξ ^ ⟨ ξ ^ ⟩ (: hat(xi):)\langle\hat{\xi}\rangle⟨ξ^⟩ of T ^ T ^ hat(T)\hat{\mathbb{T}}T^. The GIT quotient X = Y / / ξ ^ X = Y / / ⟨ ξ ^ ⟩ X=Y////(: hat(xi):)X=Y / /\langle\hat{\xi}\rangleX=Y//⟨ξ^⟩ admits an orbifold structure encoded by a log Fano pair ( X , D ) ( X , D ) (X,D)(X, D)(X,D). A straightforward calculation shows that a quasiregular ( Y , ξ ^ ) ( Y , ξ ^ ) (Y, hat(xi))(Y, \hat{\xi})(Y,ξ^) admits a Ricci-flat Kähler cone metric if and only if ( X , D ) ( X , D ) (X,D)(X, D)(X,D) admits a Kähler-Einstein metric.
In general, there are many irregular Ricci-flat Kähler cone metrics, i.e., with ξ ^ ξ ^ ∈ hat(xi)in\hat{\xi} \inξ^∈ N ^ R N ^ Q N ^ R ∖ N ^ Q hat(N)_(R)\\ hat(N)_(Q)\hat{N}_{\mathbb{R}} \backslash \hat{N}_{\mathbb{Q}}N^R∖N^Q. Recent works by Apostolov-Calderbank-Jubert-Lahdili establish an equivalence between Ricci-flat Kähler cone metrics and special g g ggg-soliton metrics. More precisely, fix any χ ^ N ^ Q + χ ^ ∈ N ^ Q + hat(chi)in hat(N)_(Q)^(+)\hat{\chi} \in \hat{N}_{\mathbb{Q}}^{+}χ^∈N^Q+and consider the quotient ( X , D ) = Y / / χ ^ ( X , D ) = Y / / ⟨ χ ^ ⟩ (X,D)=Y////(: hat(chi):)(X, D)=Y / /\langle\hat{\chi}\rangle(X,D)=Y//⟨χ^⟩ as above. It is shown in [2] (see also [47]) that the Ricci-flat Kähler cone metric on ( Y , ξ ^ ) ( Y , ξ ^ ) (Y, hat(xi))(Y, \hat{\xi})(Y,ξ^) is equivalent to the g g ggg-soliton metric on ( X , D ) ( X , D ) (X,D)(X, D)(X,D) with g ( y ) = ( n + 1 + y , ξ ) n 2 g ( y ) = ( n + 1 + ⟨ y , ξ ⟩ ) − n − 2 g(y)=(n+1+(:y,xi:))^(-n-2)g(y)=(n+1+\langle y, \xi\rangle)^{-n-2}g(y)=(n+1+⟨y,ξ⟩)−n−2 where ξ ξ xi\xiξ (equivalently, V ξ ) V ξ {:V_(xi))\left.V_{\xi}\right)Vξ) is induced by ξ ^ ξ ^ hat(xi)\hat{\xi}ξ^ on X X XXX.

1.5. Analytic criteria for the existence

We now review a well-understood criterion for the existence of above canonical Kähler metrics. The general idea is to view corresponding equations as Euler-Lagrange equations of appropriate energy functionals and then use a variational approach to prove that the existence of solutions is equivalent to the coercivity of the energy functionals. First we have the following functionals defined for any φ H φ ∈ H varphi inH\varphi \in \mathscr{H}φ∈H (see (1.1)):
(1.6) E ( φ ) = 1 ( n + 1 ) V k = 0 n X φ ω φ k ω 0 n k , Λ ( φ ) = 1 V X φ ω 0 n (1.7) J ( φ ) = Λ ( φ ) E ( φ ) , E χ ( φ ) = 1 V k = 0 n 1 X φ χ ω φ k ω n 1 k (1.6) E ( φ ) = 1 ( n + 1 ) V ∑ k = 0 n   ∫ X   φ ω φ k ∧ ω 0 n − k , Λ ( φ ) = 1 V ∫ X   φ ω 0 n (1.7) J ( φ ) = Λ ( φ ) − E ( φ ) , E χ ( φ ) = 1 V ∑ k = 0 n − 1   ∫ X   φ χ ∧ ω φ k ∧ ω n − 1 − k {:[(1.6)E(varphi)=(1)/((n+1)V)sum_(k=0)^(n)int_(X)varphiomega_(varphi)^(k)^^omega_(0)^(n-k)","quad Lambda(varphi)=(1)/(V)int_(X)varphiomega_(0)^(n)],[(1.7)J(varphi)=Lambda(varphi)-E(varphi)","quadE^(chi)(varphi)=(1)/(V)sum_(k=0)^(n-1)int_(X)varphi chi^^omega_(varphi)^(k)^^omega^(n-1-k)]:}\begin{align*} & \mathbf{E}(\varphi)=\frac{1}{(n+1) \mathbf{V}} \sum_{k=0}^{n} \int_{X} \varphi \omega_{\varphi}^{k} \wedge \omega_{0}^{n-k}, \quad \boldsymbol{\Lambda}(\varphi)=\frac{1}{\mathbf{V}} \int_{X} \varphi \omega_{0}^{n} \tag{1.6}\\ & \mathbf{J}(\varphi)=\boldsymbol{\Lambda}(\varphi)-\mathbf{E}(\varphi), \quad \mathbf{E}^{\chi}(\varphi)=\frac{1}{\mathbf{V}} \sum_{k=0}^{n-1} \int_{X} \varphi \chi \wedge \omega_{\varphi}^{k} \wedge \omega^{n-1-k} \tag{1.7} \end{align*}(1.6)E(φ)=1(n+1)V∑k=0n∫Xφωφk∧ω0n−k,Λ(φ)=1V∫Xφω0n(1.7)J(φ)=Λ(φ)−E(φ),Eχ(φ)=1V∑k=0n−1∫Xφχ∧ωφk∧ωn−1−k
Here V V V\mathbf{V}V is defined in (1.2) and χ χ chi\chiχ is any closed real (1,1)-form.
The following functionals are important for studying the cscK problem:
(1.8) H ( φ ) = 1 V X log ω φ n Ω 0 ω φ n , M ( φ ) = H ( φ ) + E Ric ( ω 0 ) ( φ ) + S _ E ( φ ) (1.8) H ( φ ) = 1 V ∫ X   log ⁡ ω φ n Ω 0 ω φ n , M ( φ ) = H ( φ ) + E − Ric ⁡ ω 0 ( φ ) + S _ â‹… E ( φ ) {:(1.8)H(varphi)=(1)/(V)int_(X)log ((omega_(varphi)^(n))/(Omega_(0))omega_(varphi)^(n))","quadM(varphi)=H(varphi)+E^(-Ric(omega_(0)))(varphi)+S_*E(varphi):}\begin{equation*} \mathbf{H}(\varphi)=\frac{1}{\mathbf{V}} \int_{X} \log \frac{\omega_{\varphi}^{n}}{\Omega_{0}} \omega_{\varphi}^{n}, \quad \mathbf{M}(\varphi)=\mathbf{H}(\varphi)+\mathbf{E}^{-\operatorname{Ric}\left(\omega_{0}\right)}(\varphi)+\underline{S} \cdot \mathbf{E}(\varphi) \tag{1.8} \end{equation*}(1.8)H(φ)=1V∫Xlog⁡ωφnΩ0ωφn,M(φ)=H(φ)+E−Ric⁡(ω0)(φ)+S_â‹…E(φ)
The above H ( φ ) H ( φ ) H(varphi)\mathbf{H}(\varphi)H(φ) is usually called the entropy of the measure ω φ n ω φ n omega_(varphi)^(n)\omega_{\varphi}^{n}ωφn. One can verify that any critical point of M M M\mathbf{M}M is the potential of a cscK metric.
For Kähler-Einstein (KE) metrics on Fano manifolds, we have more functionals:
(1.9) L ( φ ) = log ( 1 V X e φ Ω 0 ) , D ( φ ) = E ( φ ) + L ( φ ) (1.9) L ( φ ) = − log ⁡ 1 V ∫ X   e − φ Ω 0 , D ( φ ) = − E ( φ ) + L ( φ ) {:(1.9)L(varphi)=-log((1)/(V)int_(X)e^(-varphi)Omega_(0))","quadD(varphi)=-E(varphi)+L(varphi):}\begin{equation*} \mathbf{L}(\varphi)=-\log \left(\frac{1}{\mathbf{V}} \int_{X} e^{-\varphi} \Omega_{0}\right), \quad \mathbf{D}(\varphi)=-\mathbf{E}(\varphi)+\mathbf{L}(\varphi) \tag{1.9} \end{equation*}(1.9)L(φ)=−log⁡(1V∫Xe−φΩ0),D(φ)=−E(φ)+L(φ)
A critical point of D D D\mathbf{D}D is also a K E K E KE\mathrm{KE}KE potential. These functionals can be generalized to the settings of g g ggg-weighted solitons and Ricci-flat Kähler cone metrics (see [47] for references).
To apply the variational approach, one first needs a "completion" of H H H\mathscr{H}H. Such a completion was defined by Guedj-Zeriahi extending the local study of Cegrell. Following [7], one way to introduce this is to first define the E E E\mathbf{E}E functional for any φ PSH ( ω 0 ) φ ∈ PSH ⁡ ω 0 varphi in PSH(omega_(0))\varphi \in \operatorname{PSH}\left(\omega_{0}\right)φ∈PSH⁡(ω0) by
(1.10) E ( φ ) = inf { E ( φ ~ ) : φ ~ φ , φ ~ H ( ω 0 ) } (1.10) E ( φ ) = inf E ( φ ~ ) : φ ~ ≥ φ , φ ~ ∈ H ω 0 {:(1.10)E(varphi)=i n f{E(( tilde(varphi))):( tilde(varphi)) >= varphi,( tilde(varphi))inH(omega_(0))}:}\begin{equation*} \mathbf{E}(\varphi)=\inf \left\{\mathbf{E}(\tilde{\varphi}): \tilde{\varphi} \geq \varphi, \tilde{\varphi} \in \mathscr{H}\left(\omega_{0}\right)\right\} \tag{1.10} \end{equation*}(1.10)E(φ)=inf{E(φ~):φ~≥φ,φ~∈H(ω0)}
Then define the set of finite energy potentials as
(1.11) E 1 := E 1 ( ω 0 ) = { φ PSH ( ω 0 ) : E ( φ ) > } (1.11) E 1 := E 1 ω 0 = φ ∈ PSH ⁡ ω 0 : E ( φ ) > − ∞ {:(1.11)E^(1):=E^(1)(omega_(0))={varphi in PSH(omega_(0)):E(varphi) > -oo}:}\begin{equation*} \mathcal{E}^{1}:=\mathcal{E}^{1}\left(\omega_{0}\right)=\left\{\varphi \in \operatorname{PSH}\left(\omega_{0}\right): \mathbf{E}(\varphi)>-\infty\right\} \tag{1.11} \end{equation*}(1.11)E1:=E1(ω0)={φ∈PSH⁡(ω0):E(φ)>−∞}
After the work [6], ε 1 ε 1 epsi^(1)\varepsilon^{1}ε1 can be endowed with a strong topology which is the coarsest refinement of the weak topology (i.e., the L 1 L 1 L^(1)L^{1}L1-topology) that makes E E E\mathbf{E}E continuous. The above energy functionals can be extended to ε 1 ε 1 epsi^(1)\varepsilon^{1}ε1, and they satisfy important regularization properties:
Theorem 1.2 (see [ 6 , 8 ] [ 6 , 8 ] [6,8][6,8][6,8] ). For any φ E 1 φ ∈ E 1 varphi inE^(1)\varphi \in \mathcal{E}^{1}φ∈E1, there exists { φ k } k N H φ k k ∈ N ⊂ H {varphi_(k)}_(k inN)subH\left\{\varphi_{k}\right\}_{k \in \mathbb{N}} \subset \mathscr{H}{φk}k∈N⊂H such that F ( φ k ) F ( φ ) F φ k → F ( φ ) F(varphi_(k))rarrF(varphi)\mathbf{F}\left(\varphi_{k}\right) \rightarrow \mathbf{F}(\varphi)F(φk)→F(φ) for F { E , Λ , E Ric , H } F ∈ E , Λ , E − Ric  , H Fin{E,Lambda,E^(-"Ric "),H}\mathbf{F} \in\left\{\mathbf{E}, \boldsymbol{\Lambda}, \mathbf{E}^{- \text {Ric }}, \mathbf{H}\right\}F∈{E,Λ,E−Ric ,H}.
We would like to emphasize the result for F = H F = H F=H\mathbf{F}=\mathbf{H}F=H, which was proved in [8]. The idea of proof there is to first regularize the measure ω φ n ω φ n omega_(varphi)^(n)\omega_{\varphi}^{n}ωφn with converging entropy and then use Yau's solution to complex Monge-Ampère equations with prescribed volume forms. Later we will encounter the same idea in the non-Archimedean setting.
Another key concept is the geodesic between two finite energy potentials. For φ i E 1 , i = 0 , 1 φ i ∈ E 1 , i = 0 , 1 varphi_(i)inE^(1),i=0,1\varphi_{i} \in \mathcal{E}^{1}, i=0,1φi∈E1,i=0,1, the geodesic connecting them is the following p 1 ω 0 p 1 ∗ ω 0 p_(1)^(**)omega_(0)p_{1}^{*} \omega_{0}p1∗ω0-psh function on X × [ 0 , 1 ] × S 1 X × [ 0 , 1 ] × S 1 X xx[0,1]xxS^(1)X \times[0,1] \times S^{1}X×[0,1]×S1 where p 1 p 1 p_(1)p_{1}p1 is the projection to the first factor (see [ 7 , 26 ] [ 7 , 26 ] [7,26][7,26][7,26] ):
(1.12) Φ = sup { Ψ : Ψ is S 1 -invariant and p 1 ω 0 -psh, lim s i Ψ ( , s ) φ ( i ) , i = 0 , 1 } (1.12) Φ = sup Ψ : Ψ  is  S 1 -invariant and  p 1 ∗ ω 0 -psh,  lim s → i   Ψ ( â‹… , s ) ≤ φ ( i ) , i = 0 , 1 {:(1.12)Phi=s u p{Psi:Psi" is "S^(1)"-invariant and "p_(1)^(**)omega_(0)"-psh, "lim_(s rarr i)Psi(*,s) <= varphi(i),i=0,1}:}\begin{equation*} \Phi=\sup \left\{\Psi: \Psi \text { is } S^{1} \text {-invariant and } p_{1}^{*} \omega_{0} \text {-psh, } \lim _{s \rightarrow i} \Psi(\cdot, s) \leq \varphi(i), i=0,1\right\} \tag{1.12} \end{equation*}(1.12)Φ=sup{Ψ:Ψ is S1-invariant and p1∗ω0-psh, lims→iΨ(â‹…,s)≤φ(i),i=0,1}
The concept of geodesics originates from Mabuchi's L 2 L 2 L^(2)L^{2}L2-Riemannian metric on H H H\mathscr{H}H. According to the work of Semmes and Donaldson, if φ i H , i = 0 , 1 φ i ∈ H , i = 0 , 1 varphi_(i)inH,i=0,1\varphi_{i} \in \mathscr{H}, i=0,1φi∈H,i=0,1, then the geodesic Φ Î¦ Phi\PhiΦ is a solution to the Dirichlet problem of homogeneous complex Monge-Ampère equation
(1.13) ( p 1 ω 0 + dd c Φ ) n + 1 = 0 , Φ ( , i ) = φ i , i = 0 , 1 (1.13) p 1 ∗ ω 0 + dd c ⁡ Φ n + 1 = 0 , Φ ( â‹… , i ) = φ i , i = 0 , 1 {:(1.13)(p_(1)^(**)omega_(0)+dd^(c)Phi)^(n+1)=0","quad Phi(*","i)=varphi_(i)","i=0","1:}\begin{equation*} \left(p_{1}^{*} \omega_{0}+\operatorname{dd}^{\mathrm{c}} \Phi\right)^{n+1}=0, \quad \Phi(\cdot, i)=\varphi_{i}, i=0,1 \tag{1.13} \end{equation*}(1.13)(p1∗ω0+ddc⁡Φ)n+1=0,Φ(â‹…,i)=φi,i=0,1
Since Φ Î¦ Phi\PhiΦ is S 1 S 1 S^(1)S^{1}S1-invariant, we can consider Φ Î¦ Phi\PhiΦ as a family of ω 0 ω 0 omega_(0)\omega_{0}ω0-psh functions { φ ( s ) } s [ 0 , 1 ] { φ ( s ) } s ∈ [ 0 , 1 ] {varphi(s)}_(s in[0,1])\{\varphi(s)\}_{s \in[0,1]}{φ(s)}s∈[0,1].
Theorem 1.3 ( [ 5 , 8 ] ) [ 5 , 8 ] ) [5,8])[5,8])[5,8]). Let Φ = { φ ( s ) } s [ 0 , 1 ] Φ = { φ ( s ) } s ∈ [ 0 , 1 ] Phi={varphi(s)}_(s in[0,1])\Phi=\{\varphi(s)\}_{s \in[0,1]}Φ={φ(s)}s∈[0,1] be a geodesic segment in E 1 E 1 E^(1)\mathcal{E}^{1}E1. Then (1) s s ↦ s|->s \mapstos↦ E ( φ ( s ) ) E ( φ ( s ) ) E(varphi(s))\mathbf{E}(\varphi(s))E(φ(s)) is affine; (2) s M ( φ ( s ) ) s ↦ M ( φ ( s ) ) s|->M(varphi(s))s \mapsto \mathbf{M}(\varphi(s))s↦M(φ(s)) is convex.
Results in Theorem 1.3 are important in the variational approach. If a geodesic is smooth, the statements follow from straightforward calculations. However, there are examples (first due to Lempert-Vivas) showing that the solution to (1.13) in general does not have sufficient regularity. So the proofs of the above results are more involved.
In this paper T ~ T ~ tilde(T)\tilde{\mathbb{T}}T~ will always denote a maximal torus of the linear algebraic group Aut ( X , L ) Aut ⁡ ( X , L ) Aut(X,L)\operatorname{Aut}(X, L)Aut⁡(X,L) and T ~ T ~ tilde(T)\tilde{T}T~ is a maximal real subtorus of T ~ T ~ tilde(T)\tilde{\mathbb{T}}T~. In the following result, we use the translation invariance F ( φ + c ) = F ( φ ) F ( φ + c ) = F ( φ ) F(varphi+c)=F(varphi)\mathbf{F}(\varphi+c)=\mathbf{F}(\varphi)F(φ+c)=F(φ) for F { M , J } F ∈ { M , J } Fin{M,J}\mathbf{F} \in\{\mathbf{M}, \mathbf{J}\}F∈{M,J} and hence F ( ω φ ) := F ( φ ) F ω φ := F ( φ ) F(omega_(varphi)):=F(varphi)\mathbf{F}\left(\omega_{\varphi}\right):=\mathbf{F}(\varphi)F(ωφ):=F(φ) is well defined.
Theorem 1.4 ([9,23,27]). There exists a T ~ T ~ tilde(T)\tilde{T}T~-invariant cscK metric in c 1 ( L ) c 1 ( L ) c_(1)(L)c_{1}(L)c1(L) if and only if M M M\mathbf{M}M is reduced coercive, which means that there exist γ , C > 0 γ , C > 0 gamma,C > 0\gamma, C>0γ,C>0 such that for any φ H T ~ φ ∈ H T ~ varphi inH^( tilde(T))\varphi \in \mathscr{H}^{\tilde{T}}φ∈HT~,
(1.14) M ( ω φ ) γ inf σ T J ( σ ω φ ) C (1.14) M ω φ ≥ γ â‹… inf σ ∈ T   J σ ∗ ω φ − C {:(1.14)M(omega_(varphi)) >= gamma*i n f_(sigma inT)J(sigma^(**)omega_(varphi))-C:}\begin{equation*} \mathbf{M}\left(\omega_{\varphi}\right) \geq \gamma \cdot \inf _{\sigma \in \mathbb{T}} \mathbf{J}\left(\sigma^{*} \omega_{\varphi}\right)-C \tag{1.14} \end{equation*}(1.14)M(ωφ)≥γ⋅infσ∈TJ(σ∗ωφ)−C
This type of result goes back to Tian's pioneering work in [64] which proves that if X X XXX is a Fano manifold with a discrete automorphism group, then the existence of KählerEinstein metric is equivalent to the properness of the M M M\mathbf{M}M-functional, and is also equivalent to the properness of the D D D\mathbf{D}D functional. Tian's work has since been refined and generalized for other canonical metrics. For the necessity direction (from existence to reduced coercivity), there is now a general principle due to Darvas-Rubinstein ([27]) that can be applied for all previously-mentioned canonical Kähler metrics. The sufficient direction (from reduced coercivity to existence) for Kähler-Einstein metrics is reproved in [6] using pluripotential theory, which works equally well in the setting of log log log\loglog Fano pairs. See [ 10 , 38 ] [ 10 , 38 ] [10,38][10,38][10,38] for the extension to the g g ggg-soliton case. The existence result for smooth cscK metrics is accomplished recently by Chen-Cheng's new estimates [23]. The use of maximal torus appears in [44,45], refining an earlier formulation of Hisamoto [39]. There is also an existence criterion when T ~ T ~ tilde(T)\tilde{\mathbb{T}}T~ is replaced by any connected reductive subgroup of Aut ( X , L ) Aut ⁡ ( X , L ) Aut(X,L)\operatorname{Aut}(X, L)Aut⁡(X,L) that contains a maximal torus.

2. STABILITY OF ALGEBRAIC VARIETIES AND NON-ARCHIMEDEAN GEOMETRY

2.1. K-stability and non-Archimedean geometry

The concept of K-stability, as first introduced by Tian [64] and Donaldson [30], is motivated by results from geometric analysis. On the other hand, the recent development shows that various tools from algebraic geometry are crucial in unlocking many of its mysteries.
Definition 2.1. A test configuration for a polarized manifold ( X , L ) ( X , L ) (X,L)(X, L)(X,L) consists of ( X , L ) ( X , L ) (X,L)(\mathcal{X}, \mathscr{L})(X,L) that satisfies: (i) π : X C Ï€ : X → C pi:XrarrC\pi: \mathcal{X} \rightarrow \mathbb{C}Ï€:X→C is a flat projective morphism from a normal variety X X X\mathcal{X}X, and L L L\mathscr{L}L is a π Ï€ pi\piÏ€ semiample Q Q Q\mathbb{Q}Q-line bundle; (ii) There is a C C ∗ C^(**)\mathbb{C}^{*}C∗-action on ( X , L ) ( X , L ) (X,L)(\mathcal{X}, \mathscr{L})(X,L) such that π Ï€ pi\piÏ€ is C C ∗ C^(**)\mathbb{C}^{*}C∗-equivariant; (iii) There is a C C ∗ C^(**)\mathbb{C}^{*}C∗-equivariant isomorphism ( X , L ) × C C ( X × C , p 1 L ) ( X , L ) × C C ∗ ≅ X × C ∗ , p 1 ∗ L (X,L)xx_(C)C^(**)~=(X xxC^(**),p_(1)^(**)L)(\mathcal{X}, \mathscr{L}) \times_{\mathbb{C}} \mathbb{C}^{*} \cong\left(X \times \mathbb{C}^{*}, p_{1}^{*} L\right)(X,L)×CC∗≅(X×C∗,p1∗L).
Configuration ( X , L ) ( X , L ) (X,L)(\mathcal{X}, \mathscr{L})(X,L) is called a product test configuration if there is a C C ∗ C^(**)\mathbb{C}^{*}C∗-equivariant isomorphism ( X , L ) ( X × C , p 1 L ) ( X , L ) ≅ X × C , p 1 ∗ L (X,L)~=(X xxC,p_(1)^(**)L)(\mathcal{X}, \mathscr{L}) \cong\left(X \times \mathbb{C}, p_{1}^{*} L\right)(X,L)≅(X×C,p1∗L) where the C C ∗ C^(**)\mathbb{C}^{*}C∗-action on the right-hand side is the product action of a C C ∗ C^(**)\mathbb{C}^{*}C∗-action on ( X , L ) ( X , L ) (X,L)(X, L)(X,L) with the standard multiplication on C C C\mathbb{C}C.
Two test configurations ( X i , L i ) , i = 1 , 2 X i , L i , i = 1 , 2 (X_(i),L_(i)),i=1,2\left(\mathcal{X}_{i}, \mathscr{L}_{i}\right), i=1,2(Xi,Li),i=1,2 are called equivalent if there exists a test configuration ( X , L ) X ′ , L ′ (X^('),L^('))\left(\mathcal{X}^{\prime}, \mathscr{L}^{\prime}\right)(X′,L′) with C C ∗ C^(**)\mathbb{C}^{*}C∗-equivariant birational morphisms ρ i : X X i ρ i : X ′ → X i rho_(i):X^(')rarrX_(i)\rho_{i}: X^{\prime} \rightarrow X_{i}ρi:X′→Xi satisfying ρ 1 L 1 = L = ρ 2 L 2 ρ 1 ∗ L 1 = L ′ = ρ 2 ∗ L 2 rho_(1)^(**)L_(1)=L^(')=rho_(2)^(**)L_(2)\rho_{1}^{*} \mathscr{L}_{1}=\mathscr{L}^{\prime}=\rho_{2}^{*} \mathscr{L}_{2}ρ1∗L1=L′=ρ2∗L2. For any test configuration ( X , L ) ( X , L ) (X,L)(\mathcal{X}, \mathscr{L})(X,L), by taking fiber product, one can always find an equivalent test configuration ( X , L ) X ′ , L ′ (X^('),L^('))\left(\mathcal{X}^{\prime}, \mathscr{L}^{\prime}\right)(X′,L′) such that X X ′ X^(')\mathcal{X}^{\prime}X′ dominates X × C X × C X xxCX \times \mathbb{C}X×C.
Given any test configuration ( X , L ) ( X , L ) (X,L)(\mathcal{X}, \mathscr{L})(X,L), there is a canonical compactification over P 1 P 1 P^(1)\mathbb{P}^{1}P1 denoted by ( X ¯ , L ¯ ) ( X ¯ , L ¯ ) ( bar(X), bar(L))(\overline{\mathcal{X}}, \overline{\mathscr{L}})(X¯,L¯) which is obtained by adding a trivial fiber over { } = P 1 C { ∞ } = P 1 ∖ C {oo}=P^(1)\\C\{\infty\}=\mathbb{P}^{1} \backslash \mathbb{C}{∞}=P1∖C.
The notion of a test configuration is a way to formulate the degeneration of ( X , L ) ( X , L ) (X,L)(X, L)(X,L). In fact, any test configuration is induced by a one-parameter subgroup of PGL ( N + 1 , C ) PGL ⁡ ( N + 1 , C ) PGL(N+1,C)\operatorname{PGL}(N+1, \mathbb{C})PGL⁡(N+1,C) for a Kodaira embedding X P N X → P N X rarrP^(N)X \rightarrow \mathbb{P}^{N}X→PN.
We will continue our discussion in a framework of non-Archimedean geometry as proposed by Boucksom-Jonsson. Let X N A X N A X^(NA)X^{\mathrm{NA}}XNA denote the Berkovich analytification of X X XXX with respect to the trivial absolute value on C C C\mathbb{C}C (see [18] for references). X N A X N A X^(NA)X^{\mathrm{NA}}XNA is a topological space consisting of real valuations on subvarieties of X X XXX, and contains a dense subset X Q div X Q div  X_(Q)^("div ")X_{\mathbb{Q}}^{\text {div }}XQdiv  consisting of divisorial valuations on X X XXX. Any test configuration ( X , L ) ( X , L ) (X,L)(\mathcal{X}, \mathscr{L})(X,L) defines a function on X N A X N A X^(NA)X^{\mathrm{NA}}XNA in the following way. First, up to equivalence, we can assume that there is a birational morphism ρ : X X C := X × C ρ : X → X C := X × C rho:XrarrX_(C):=X xxC\rho: \mathcal{X} \rightarrow X_{\mathbb{C}}:=X \times \mathbb{C}ρ:X→XC:=X×C. Write L = ρ p 1 L + E L = ρ ∗ p 1 ∗ L + E L=rho^(**)p_(1)^(**)L+E\mathscr{L}=\rho^{*} p_{1}^{*} L+EL=ρ∗p1∗L+E where E E EEE is a Q Q Q\mathbb{Q}Q-divisor supported on X 0 X 0 X_(0)X_{0}X0. For any v X N A v ∈ X N A v inX^(NA)v \in X^{\mathrm{NA}}v∈XNA, denote by G ( v ) G ( v ) G(v)G(v)G(v) the C C ∗ C^(**)\mathbb{C}^{*}C∗-invariant semivaluation on X C X C X_(C)X_{\mathbb{C}}XC that satisfies G ( v ) | C ( X ) = v G ( v ) C ( X ) = v G(v)|_(C(X))=v\left.G(v)\right|_{\mathbb{C}(X)}=vG(v)|C(X)=v and G ( v ) ( t ) = 1 G ( v ) ( t ) = 1 G(v)(t)=1G(v)(t)=1G(v)(t)=1 where t t ttt is the coordinate of C C C\mathbb{C}C. One then defines
(2.1) ϕ ( X , L ) ( v ) = G ( v ) ( E ) , for any v X N A (2.1) Ï• ( X , L ) ( v ) = G ( v ) ( E ) ,  for any  v ∈ X N A {:(2.1)phi_((X,L))(v)=G(v)(E)","quad" for any "v inX^(NA):}\begin{equation*} \phi_{(X, \mathscr{L})}(v)=G(v)(E), \quad \text { for any } v \in X^{\mathrm{NA}} \tag{2.1} \end{equation*}(2.1)Ï•(X,L)(v)=G(v)(E), for any v∈XNA
The set of such functions on X N A X N A X^(NA)X^{\mathrm{NA}}XNA obtained from test configurations is denoted by H N A H N A H^(NA)\mathscr{H}^{\mathrm{NA}}HNA which is considered as the set of smooth non-Archimedean psh potentials on the analytification of L L LLL. The following functionals, defined on the space of test configurations, correspond to the Archimedean (i.e., complex-analytic) functionals in (1.6)-(1.7):
(2.2) E N A ( X , L ) = L ¯ n + 1 ( n + 1 ) V , Λ N A ( X , L ) = 1 V L ¯ n ρ L P 1 (2.3) J N A ( X , L ) = Λ N A ( X , L ) E N A ( X , L ) , ( E K X ) N A ( X , L ) = 1 V K X L ¯ n (2.4) H N A ( X , L ) = 1 V K X ¯ / X P 1 log L n , M N A ( X , L ) = H N A + ( E K X ) N A + S _ E N A (2.2) E N A ( X , L ) = L ¯ â‹… n + 1 ( n + 1 ) V , Λ N A ( X , L ) = 1 V L ¯ n â‹… ρ ∗ L P 1 (2.3) J N A ( X , L ) = Λ N A ( X , L ) − E N A ( X , L ) , E K X N A ( X , L ) = 1 V K X â‹… L ¯ n (2.4) H N A ( X , L ) = 1 V K X ¯ / X P 1 log â‹… L â‹… n , M N A ( X , L ) = H N A + E K X N A + S _ â‹… E N A {:[(2.2)E^(NA)(X","L)=( bar(L)*n+1)/((n+1)V)","quadLambda^(NA)(X","L)=(1)/(V) bar(L)^(n)*rho^(**)L_(P^(1))],[(2.3)J^(NA)(X","L)=Lambda^(NA)(X","L)-E^(NA)(X","L)","quad(E^(K_(X)))^(NA)(X","L)=(1)/(V)K_(X)* bar(L)^(n)],[(2.4)H^(NA)(X","L)=(1)/(V)K_( bar(X)//X_(P^(1)))^(log)*L^(*n)","quadM^(NA)(X","L)=H^(NA)+(E^(K_(X)))^(NA)+S_*E^(NA)]:}\begin{align*} & \mathbf{E}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})=\frac{\overline{\mathscr{L}} \cdot n+1}{(n+1) \mathbf{V}}, \quad \Lambda^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})=\frac{1}{\mathbf{V}} \bar{L}^{n} \cdot \rho^{*} L_{\mathbb{P}^{1}} \tag{2.2}\\ & \mathbf{J}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})=\boldsymbol{\Lambda}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})-\mathbf{E}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L}), \quad\left(\mathbf{E}^{K_{X}}\right)^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})=\frac{1}{\mathbf{V}} K_{X} \cdot \bar{L}^{n} \tag{2.3}\\ & \mathbf{H}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})=\frac{1}{\mathbf{V}} K_{\bar{X} / X_{\mathbb{P}^{1}}}^{\log } \cdot \mathscr{L}^{\cdot n}, \quad \mathbf{M}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})=\mathbf{H}^{\mathrm{NA}}+\left(\mathbf{E}^{K_{X}}\right)^{\mathrm{NA}}+\underline{S} \cdot \mathbf{E}^{\mathrm{NA}} \tag{2.4} \end{align*}(2.2)ENA(X,L)=L¯⋅n+1(n+1)V,ΛNA(X,L)=1VL¯n⋅ρ∗LP1(2.3)JNA(X,L)=ΛNA(X,L)−ENA(X,L),(EKX)NA(X,L)=1VKXâ‹…L¯n(2.4)HNA(X,L)=1VKX¯/XP1logâ‹…Lâ‹…n,MNA(X,L)=HNA+(EKX)NA+S_â‹…ENA
where we assume that X ¯ X ¯ bar(X)\bar{X}X¯ dominates X P 1 = X × P 1 X P 1 = X × P 1 X_(P^(1))=X xxP^(1)X_{\mathbb{P}^{1}}=X \times \mathbb{P}^{1}XP1=X×P1 by ρ ρ rho\rhoρ, and L P 1 = p 1 L , K x ¯ / X P 1 10 log = K x ¯ + L P 1 = p 1 ∗ L , K x ¯ / X P 1 10 log = K x ¯ + L_(P^(1))=p_(1)^(**)L,K_( bar(x)//X_(P1)^(10))^(log)=K bar(x)+L_{\mathbb{P}^{1}}=p_{1}^{*} L, K_{\bar{x} / X_{\mathbb{P} 1}^{10}}^{\log }=K \bar{x}+LP1=p1∗L,Kx¯/XP110log=Kx¯+ X 0 r e d ( ρ ( K X × P 1 + X × { 0 } ) ) X 0 r e d − ρ ∗ K X × P 1 + X × { 0 } X_(0)^(red)-(rho^(**)(K_(X xxP^(1))+X xx{0}))X_{0}^{\mathrm{red}}-\left(\rho^{*}\left(K_{X \times \mathbb{P}^{1}}+X \times\{0\}\right)\right)X0red−(ρ∗(KX×P1+X×{0})). These functionals were defined before the introduction of the non-Archimedean framework. For example, the E N A E N A E^(NA)\mathbf{E}^{\mathrm{NA}}ENA functional appeared in Mumford's study of Chow stability of projective varieties.
Assume that X 0 = i b i F i X 0 = ∑ i   b i F i X_(0)=sum_(i)b_(i)F_(i)\mathcal{X}_{0}=\sum_{i} b_{i} F_{i}X0=∑ibiFi where F i F i F_(i)F_{i}Fi are irreducible components. Set v i = v i = v_(i)=v_{i}=vi= b i 1 ord F i p 1 X Q div b i − 1 ord F i ∘ p 1 ∗ ∈ X Q div  b_(i)^(-1)ord_(F_(i))@p_(1)^(**)inX_(Q)^("div ")b_{i}^{-1} \operatorname{ord}_{F_{i}} \circ p_{1}^{*} \in X_{\mathbb{Q}}^{\text {div }}bi−1ordFi∘p1∗∈XQdiv  and let δ v i δ v i delta_(v_(i))\delta_{v_{i}}δvi be the Dirac measure supported at { v i } v i {v_(i)}\left\{v_{i}\right\}{vi}. Chambert-Loir defined the following non-Archimedean Monge-Ampère measure using the intersection theory:
(2.5) MA N A ( ϕ ( x , L ) ) = i b i ( L n F i ) δ v i (2.5) MA N A ⁡ Ï• ( x , L ) = ∑ i   b i L â‹… n â‹… F i δ v i {:(2.5)MA^(NA)(phi_((x,L)))=sum_(i)b_(i)(L^(*n)*F_(i))delta_(v_(i)):}\begin{equation*} \operatorname{MA}^{\mathrm{NA}}\left(\phi_{(x, \mathscr{L})}\right)=\sum_{i} b_{i}\left(\mathscr{L}^{\cdot n} \cdot F_{i}\right) \delta_{v_{i}} \tag{2.5} \end{equation*}(2.5)MANA⁡(Ï•(x,L))=∑ibi(Lâ‹…nâ‹…Fi)δvi
Mixed non-Archimedean Monge-Ampère measures are similarly defined. It then turns out that the functionals from (2.2)-(2.3) can be obtained by using the same formula as in (1.6)-(1.7) but with the ordinary integrals replaced by corresponding non-Archimedean ones, while the H N A H N A H^(NA)\mathbf{H}^{\mathrm{NA}}HNA functional has the following expression after [19]:
(2.6) H N A ( X , L ) = 1 V X N A A X ( v ) M A N A ( ϕ ( x , L ) ) ( v ) (2.6) H N A ( X , L ) = 1 V ∫ X N A   A X ( v ) M A N A Ï• ( x , L ) ( v ) {:(2.6)H^(NA)(X","L)=(1)/(V)int_(X^(NA))A_(X)(v)MA^(NA)(phi_((x,L)))(v):}\begin{equation*} \mathbf{H}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})=\frac{1}{\mathbf{V}} \int_{X^{\mathrm{NA}}} A_{X}(v) \mathrm{MA}^{\mathrm{NA}}\left(\phi_{(x, \mathscr{L})}\right)(v) \tag{2.6} \end{equation*}(2.6)HNA(X,L)=1V∫XNAAX(v)MANA(Ï•(x,L))(v)
Here A X A X A_(X)A_{X}AX is a functional defined on X N A X N A X^(NA)X^{\mathrm{NA}}XNA that generalizes the log discrepancy functional on X Q div X Q div  X_(Q)^("div ")X_{\mathbb{Q}}^{\text {div }}XQdiv  (see [41]). We can now recall the notion of K-stability:
Definition 2.2. A polarized manifold ( X , L ) ( X , L ) (X,L)(X, L)(X,L) is K-semistable, K-stable or K-polystable if any nontrivial test configuration ( X , L ) ( X , L ) (X,L)(\mathcal{X}, \mathscr{L})(X,L) for ( X , L ) ( X , L ) (X,L)(X, L)(X,L) satisfies M N A ( X , L ) 0 , M N A ( X , L ) > 0 M N A ( X , L ) ≥ 0 , M N A ( X , L ) > 0 M^(NA)(X,L) >= 0,M^(NA)(X,L) > 0\mathbf{M}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L}) \geq 0, \mathbf{M}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L})>0MNA(X,L)≥0,MNA(X,L)>0, or M N A ( X , L ) 0 M N A ( X , L ) ≥ 0 M^(NA)(X,L) >= 0\mathbf{M}^{\mathrm{NA}}(\mathcal{X}, \mathscr{L}) \geq 0MNA(X,L)≥0 and = 0 = 0 =0=0=0 only if ( X , L ) ( X , L ) (X,L)(\mathcal{X}, \mathscr{L})(X,L) is a product test configuration, respectively.
This is like a Hilbert-Mumford's numerical criterion in the Geometric Invariant Theory. 1 1 ^(1){ }^{1}1 The recent development of K K K